integral of (2^{sqrt(x)})/(sqrt(x))

{ "query": { "display": "$$\\int\\:\\frac{2^{\\sqrt{x}}}{\\sqrt{x}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{2^{\\sqrt{x}}}{\\sqrt{x}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{2\\cdot 2^{\\sqrt{x}}}{\\ln(2)}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{2^{\\sqrt{x}}}{\\sqrt{x}}dx=\\frac{2\\cdot\\:2^{\\sqrt{x}}}{\\ln\\left(2\\right)}+C$$", "input": "\\int\\:\\frac{2^{\\sqrt{x}}}{\\sqrt{x}}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{2^{\\sqrt{x}}}{\\sqrt{x}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=2^{\\sqrt{x}}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\frac{\\ln\\left(2\\right)2^{\\sqrt{x}-1}}{\\sqrt{x}}$$", "input": "\\frac{d}{dx}\\left(2^{\\sqrt{x}}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b}=e^{b\\ln\\left(a\\right)}$$", "secondary": [ "$$2^{\\sqrt{x}}=e^{\\sqrt{x}\\ln\\left(2\\right)}$$" ], "result": "=\\frac{d}{dx}\\left(e^{\\sqrt{x}\\ln\\left(2\\right)}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{\\sqrt{x}\\ln\\left(2\\right)}\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(e^{\\sqrt{x}\\ln\\left(2\\right)}\\right)", "result": "=e^{\\sqrt{x}\\ln\\left(2\\right)}\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=\\sqrt{x}\\ln\\left(2\\right)$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{d}{du}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YrHahlpzKGY893KZ4T4i4Tv3RCXWsqiNx7T9zOhL5sYfw==" } }, { "type": "step", "result": "=e^{u}\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sqrt{x}\\ln\\left(2\\right)$$", "result": "=e^{\\sqrt{x}\\ln\\left(2\\right)}\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYizAipU6viOqFOlp4SKy0GJJwhFIkm1w9WvqrAJGvIcZSUM9pakkKILvT6Fs/PM352UPWd0or+4i3R6mQ374eeZpkpAtzMPRA2gXdSSjY8vEUVEOoWGPamZg4hw64aHggTS10nwb8fz+Zyy/S88LvuHtVZAvaf5kRozrUaO4vuS2sqInyO9MO+ueRxiJUdT5EErbw/BigzSSTfKortF0XFWQGAL20SqpEVQ2gwR5EgbQ8GB5H5hx083tyV5Ny44iaQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)=\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x}\\ln\\left(2\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\ln\\left(2\\right)\\frac{d}{dx}\\left(\\sqrt{x}\\right)" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\ln\\left(2\\right)\\frac{d}{dx}\\left(x^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\ln\\left(2\\right)\\frac{1}{2}x^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\ln\\left(2\\right)\\frac{1}{2}x^{\\frac{1}{2}-1}:{\\quad}\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}$$", "input": "\\ln\\left(2\\right)\\frac{1}{2}x^{\\frac{1}{2}-1}", "result": "=\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\ln\\left(2\\right)\\frac{1\\cdot\\:x^{\\frac{1}{2}-1}}{2}" }, { "type": "interim", "title": "Simplify $$\\frac{1\\cdot\\:x^{\\frac{1}{2}-1}}{2}:{\\quad}\\frac{1}{2\\sqrt{x}}$$", "input": "\\frac{1\\cdot\\:x^{\\frac{1}{2}-1}}{2}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{\\frac{1}{2}-1}=x^{\\frac{1}{2}-1}$$", "result": "=\\frac{x^{\\frac{1}{2}-1}}{2}" }, { "type": "interim", "title": "$$x^{\\frac{1}{2}-1}=x^{-\\frac{1}{2}}$$", "input": "x^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=x^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{1}{2}-\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1-1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$1-1\\cdot\\:2=-1$$", "input": "1-1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=1-2" }, { "type": "step", "primary": "Subtract the numbers: $$1-2=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7a6Wn/M/Ttm1RnNP3N256Gd6GQqufR6tr2vPxOUv7H++mVbhxxoS+yoOibIfVJoD8phy8YAqlFyHHXjiLfa6mhzIL6DQPgBZY7Lm0+2F3hZY=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qijEBDcyPMwV4Y1jeiGyoO0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjQYCY06ctBCI/puUxKEtzAQH2kDe5DGYTz3TrPquGdIjtHZXPNLHlLyai31n5HH4G6M8osviUPEkWv33aMbZrSFQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{x^{-\\frac{1}{2}}}{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{x}}$$" ], "result": "=\\frac{\\frac{1}{\\sqrt{x}}}{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{\\sqrt{x}\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\ln\\left(2\\right)\\frac{1}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\ln\\left(2\\right)}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln\\left(2\\right)=\\ln\\left(2\\right)$$", "result": "=\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EKJZLwOZYyGhic7t+nKA1ZsiV7C57epPiXl3I8ECYg8Y8/hL79cH6K90E5uTdEyHCUCWbkwGOY7PqKo3U/JLJZAlf+65os/BiWdgfUCOm/o0xLIujH4cmDbLKYILJzFpZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz3soOz2l1BYc98G9W1IBaF/myJXsLnt6k+JeXcjwQJiD4ftFAXhJQo6Gsi4HljdwZE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{\\sqrt{x}\\ln\\left(2\\right)}\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}" }, { "type": "interim", "title": "Simplify $$e^{\\sqrt{x}\\ln\\left(2\\right)}\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}:{\\quad}\\frac{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}{\\sqrt{x}}$$", "input": "e^{\\sqrt{x}\\ln\\left(2\\right)}\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}", "result": "=\\frac{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}{\\sqrt{x}}", "steps": [ { "type": "interim", "title": "$$e^{\\sqrt{x}\\ln\\left(2\\right)}=2^{\\sqrt{x}}$$", "input": "e^{\\sqrt{x}\\ln\\left(2\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "result": "=\\left(e^{\\ln\\left(2\\right)}\\right)^{\\sqrt{x}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "secondary": [ "$$e^{\\ln\\left(2\\right)}=2$$" ], "result": "=2^{\\sqrt{x}}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nrQhzEPqwjjqfHf53jkwB12E0ali5zQi3cqbW75+rWerju+5Z51e/ZZSD3gRHwjB8FAazx82pYv5ofGe/NBYpz/L0MoYg+CUn6oyL3EO7YraUHvEkchjjzfAAncDSMGaowBh+BKH03m52JhsbUzT/BZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "step", "result": "=2^{\\sqrt{x}}\\cdot\\:\\frac{\\ln\\left(2\\right)}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}}}{2\\sqrt{x}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}\\:=\\:x^{a-b}$$", "secondary": [ "$$\\frac{2^{\\sqrt{x}}}{2}=2^{\\sqrt{x}-1}$$" ], "result": "=\\frac{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}{\\sqrt{x}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nrQhzEPqwjjqfHf53jkwB/A7GA2ILLIbBlIEkZQRg91Ccf5cKKmI0XOks6APgU8efJEe4bDv60zkwRNMOPq9UULJUwysTkf+ifmThr1HGrrqI9/58p1jQ8Hx+ZrYIL8c/BMhxhZccwF2b6nUGm8kRkD9BZ8DS96KHEpoOyRjvtCjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIKwdL24PkNyCH5w1Ln8wljBDkBaR5SPImiQgqnrmXWwyUvfrm4wN+MmqlmoA9te9S7CI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{\\ln\\left(2\\right)2^{\\sqrt{x}-1}}{\\sqrt{x}}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{\\sqrt{x}}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{u}{\\sqrt{x}}\\cdot\\:\\frac{\\sqrt{x}}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}du" }, { "type": "interim", "title": "Simplify $$\\frac{u}{\\sqrt{x}}\\cdot\\:\\frac{\\sqrt{x}}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}:{\\quad}\\frac{u}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}$$", "input": "\\frac{u}{\\sqrt{x}}\\cdot\\:\\frac{\\sqrt{x}}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{u\\sqrt{x}}{\\sqrt{x}\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{x}$$", "result": "=\\frac{u}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{u}{\\ln\\left(2\\right)\\cdot\\:2^{\\sqrt{x}-1}}du" }, { "type": "interim", "title": "$$u=2^{\\sqrt{x}}\\quad\\Rightarrow\\quad\\:\\sqrt{x}=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}$$", "input": "2^{\\sqrt{x}}=u", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "2^{\\sqrt{x}}=u", "result": "\\sqrt{x}\\ln\\left(2\\right)=\\ln\\left(u\\right)", "steps": [ { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right)$$, then $$\\ln\\left(f\\left(x\\right)\\right)=\\ln\\left(g\\left(x\\right)\\right)$$", "result": "\\ln\\left(2^{\\sqrt{x}}\\right)=\\ln\\left(u\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(x^a\\right)=a\\cdot\\ln\\left(x\\right)$$", "secondary": [ "$$\\ln\\left(2^{\\sqrt{x}}\\right)=\\sqrt{x}\\ln\\left(2\\right)$$" ], "result": "\\sqrt{x}\\ln\\left(2\\right)=\\ln\\left(u\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GF5XhT9f2I+HysjjDN8rFcxDmhxJr0J8veH7riVh222r3sVqqACdUwgkNm7Ipjz2o4xvGadv1C9e8vx7FOJ6Tl58xUv7vUkIWG/+kD/BUJkQ6hway8uxw5QgABr+PAoyv3NQC7XvhEhPr5SgcYt7oaqQhfqSx4Ip0S2H5wgufDIOPJvqJrHlzV4JQ3hSEjH25xnF+ex8t32KKJv2caSuRg==" } }, { "type": "interim", "title": "Solve $$\\sqrt{x}\\ln\\left(2\\right)=\\ln\\left(u\\right):{\\quad}\\sqrt{x}=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}$$", "input": "\\sqrt{x}\\ln\\left(2\\right)=\\ln\\left(u\\right)", "steps": [ { "type": "interim", "title": "Divide both sides by $$\\ln\\left(2\\right)$$", "input": "\\sqrt{x}\\ln\\left(2\\right)=\\ln\\left(u\\right)", "result": "\\sqrt{x}=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\ln\\left(2\\right)$$", "result": "\\frac{\\sqrt{x}\\ln\\left(2\\right)}{\\ln\\left(2\\right)}=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{x}=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\sqrt{x}=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{u}{\\ln\\left(2\\right)\\cdot\\:2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}-1}}du" }, { "type": "interim", "title": "Join $$\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}-1:{\\quad}\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}$$", "input": "\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}-1", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\ln\\left(2\\right)}{\\ln\\left(2\\right)}$$", "result": "=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}-\\frac{1\\cdot\\:\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\ln\\left(u\\right)-1\\cdot\\:\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln\\left(2\\right)=\\ln\\left(2\\right)$$", "result": "=\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{u}{\\ln\\left(2\\right)\\cdot\\:2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7ypqE1T/UtQREye/jkMLkZS/C/QhygfJVEmqjJco4hVzq/tM19MKkpgFbXOzVfC9AP+rpoTPOJyRosQJI2hFN+lsiU2T307EdVHE1M/3ZCa9+8y6OJjZj2vhn4c3a4/cyWr2JS9MJ87/lvyTUsSSZJfb5p4O9cZMZPYQ1P7mTdUNISfckYjQnTR/pBogcx4N8GaAj60mF+s+QKCRV9nRBKCBBTEk/JQ2cZ9WKuRzClU7Bi3jMVZeecTGMRH4ygWrbpXEFJTDoZKkLxfTvEPP2eY=" } }, { "type": "step", "result": "=\\int\\:\\frac{u}{\\ln\\left(2\\right)\\cdot\\:2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{\\ln\\left(2\\right)}\\cdot\\:\\int\\:\\frac{u}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}du" }, { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:\\frac{u}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}du", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}$$" }, { "type": "step", "primary": "$$v'=u$$" }, { "type": "interim", "title": "$$u'=\\frac{d}{du}\\left(\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\right)=-\\frac{2}{u^{2}}$$", "input": "\\frac{d}{du}\\left(\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{d}{du}\\left(2^{-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b}=e^{b\\ln\\left(a\\right)}$$", "secondary": [ "$$2^{-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}=e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}$$" ], "result": "=\\frac{d}{du}\\left(e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)$$", "input": "\\frac{d}{du}\\left(e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\right)", "result": "=e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{v},\\:\\:v=\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)$$" ], "result": "=\\frac{d}{dv}\\left(e^{v}\\right)\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(e^{v}\\right)=e^{v}$$", "input": "\\frac{d}{dv}\\left(e^{v}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dv}\\left(e^{v}\\right)=e^{v}$$", "result": "=e^{v}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYv5Pg3a3bzAaGV3o6+CFJwSk3hxk9aCfAWodBRxXgUexkzxTgp8EDF+A2c5wYPOrlT/L0MoYg+CUn6oyL3EO7Yq3BQfJinKm+sjzzWIgZRz2EQjeE87LX7mhk7KmLPaM8Q==" } }, { "type": "step", "result": "=e^{v}\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$v=\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)$$", "result": "=e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnmIpIfoVL0qjlGrnc5migeCBQN2cs1key3hk0/wPYgmBrQdeCfPaKiuC/e5tb+e5THIv1qnltuAAtC3ZbPnxxCo/JI5bBgpgExN510TA5cymrcR3coV6Icw5MKirx4qJ1sl9/vhh4kdzC1LX96quMx4Rlg0S4DRY7oiV7rxL/2orw+ACfJwKzIMvonzdMkBT6T9B6HUcaVUPpEYwJxD9dhPxA3JdK0T8GlsH1BnNoP/5MZzBR7/Pt4BKP5m+yHQ4wnskywhDVe4ZsNrLckpKUt/mLEmYFsNTETR0PHRpIZcUgb+os4Xi1VCkP3pqJAFMpraCFSIiPDfWV+TyTY+tciwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)=-\\frac{1}{u}$$", "input": "\\frac{d}{du}\\left(\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right):{\\quad}-\\ln\\left(u\\right)+\\ln\\left(2\\right)$$", "input": "\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\ln\\left(2\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{\\left(\\ln\\left(u\\right)-\\ln\\left(2\\right)\\right)\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\ln\\left(2\\right)$$", "result": "=-\\ln\\left(u\\right)-\\ln\\left(2\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\ln\\left(u\\right)\\right)-\\left(-\\ln\\left(2\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-\\ln\\left(u\\right)+\\ln\\left(2\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{du}\\left(-\\ln\\left(u\\right)+\\ln\\left(2\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=-\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)+\\frac{d}{du}\\left(\\ln\\left(2\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "input": "\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "result": "=\\frac{1}{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoqTCAmruKWcJsn66ZPDMT8cjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJq1PlbV5jLoKv9solFCc4blTW26qciuyUBGXQExCUedYd9mDo5FIvzrirtH7/W8pPUxk6YPA4jUd3Af4X0JJJ64=" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\ln\\left(2\\right)\\right)=0$$", "input": "\\frac{d}{du}\\left(\\ln\\left(2\\right)\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmUCfq1oxEQguNeteC39fNgcjlLRK1jUV206qo4+vRN7Bku4O63WG+AnlkHjwNvJVaN6Hv6MoTMtvtU0IQwXdn9vE00JtGmsFpd80Y1vEJJWegVCrHMbnzJchevzkQI+vw==" } }, { "type": "step", "result": "=-\\frac{1}{u}+0" }, { "type": "step", "primary": "Simplify", "result": "=-\\frac{1}{u}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\left(-\\frac{1}{u}\\right)" }, { "type": "interim", "title": "Simplify $$e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\left(-\\frac{1}{u}\\right):{\\quad}-\\frac{2}{u^{2}}$$", "input": "e^{\\left(-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\right)\\ln\\left(2\\right)}\\left(-\\frac{1}{u}\\right)", "result": "=-\\frac{2}{u^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-e^{-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\ln\\left(2\\right)}\\frac{1}{u}" }, { "type": "interim", "title": "$$e^{-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\ln\\left(2\\right)}=\\frac{2}{u}$$", "input": "e^{-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\ln\\left(2\\right)}", "steps": [ { "type": "interim", "title": "Multiply $$-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\ln\\left(2\\right)\\::{\\quad}-\\ln\\left(u\\right)+\\ln\\left(2\\right)$$", "input": "-\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}\\ln\\left(2\\right)", "result": "=e^{-\\ln\\left(u\\right)+\\ln\\left(2\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{\\left(\\ln\\left(u\\right)-\\ln\\left(2\\right)\\right)\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\ln\\left(2\\right)$$", "result": "=-\\ln\\left(u\\right)-\\ln\\left(2\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(\\ln\\left(u\\right)\\right)-\\left(-\\ln\\left(2\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-\\ln\\left(u\\right)+\\ln\\left(2\\right)" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "result": "=e^{-\\ln\\left(u\\right)}e^{\\ln\\left(2\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$e^{-\\ln\\left(u\\right)}:{\\quad}u^{-1}$$", "input": "e^{-\\ln\\left(u\\right)}", "result": "=u^{-1}e^{\\ln\\left(2\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "result": "=\\left(e^{\\ln\\left(u\\right)}\\right)^{-1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "secondary": [ "$$e^{\\ln\\left(u\\right)}=u$$" ], "result": "=u^{-1}", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$e^{\\ln\\left(2\\right)}:{\\quad}2$$", "input": "e^{\\ln\\left(2\\right)}", "result": "=u^{-1}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Apply log rule: $$a^{\\log_{a}\\left(b\\right)}=b$$", "result": "=2", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "secondary": [ "$$u^{-1}=\\frac{1}{u}$$" ], "result": "=2\\cdot\\:\\frac{1}{u}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{u}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nti66gAOYrkKa3kv1JWzoKQcbqs+3Q0Pr7Pyb4nGClgNECrFBRDPcuaFjXmI9laEVdNK6b/PmQukzNdLVQkh7NMQ0vmOWo9IZfhei7w2gJb2UVlSI3TveTGnqlyhCD5zws7amTluOikCMdcpaXqR9O0E3MJlvlo0wUi2zCNdi+dCc0mISh3+m63L8RDm9auFlUZMJJskvOvHMTY94jUsgbCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=-\\frac{2}{u}\\cdot\\:\\frac{1}{u}" }, { "type": "interim", "title": "Simplify", "input": "-\\frac{2}{u}\\cdot\\:\\frac{1}{u}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{2\\cdot\\:1}{uu}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=-\\frac{2}{uu}" }, { "type": "interim", "title": "$$uu=u^{2}$$", "input": "uu", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/E93FIYHpq26Gj2mwfeoqMzBWJotReR4P4m6RE6FZ2Oes25OoAq8kAwRqD36EFJe4ylfb0DUJOE0oSeuKQ0IOSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=-\\frac{2}{u^{2}}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } }, { "type": "step", "result": "=-\\frac{2}{u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$v=\\int\\:udu=\\frac{u^{2}}{2}$$", "input": "\\int\\:udu", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:udu", "result": "=\\frac{u^{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{1+1}}{1+1}:{\\quad}\\frac{u^{2}}{2}$$", "input": "\\frac{u^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77irOeniMfrJKKN+TrhAcvL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpAUgTzPSdH5PWV4NCtvwjA7/YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{u^{2}}{2}+C", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\cdot\\:\\frac{u^{2}}{2}-\\int\\:\\left(-\\frac{2}{u^{2}}\\right)\\frac{u^{2}}{2}du" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\cdot\\:\\frac{u^{2}}{2}-\\int\\:\\left(-\\frac{2}{u^{2}}\\right)\\frac{u^{2}}{2}du", "result": "=\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}}-\\int\\:-1du", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\cdot\\:\\frac{u^{2}}{2}-\\int\\:-\\frac{2}{u^{2}}\\cdot\\:\\frac{u^{2}}{2}du" }, { "type": "interim", "title": "$$\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\cdot\\:\\frac{u^{2}}{2}=\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}}$$", "input": "\\frac{1}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}\\cdot\\:\\frac{u^{2}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:u^{2}}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}\\cdot\\:2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:u^{2}=u^{2}$$", "result": "=\\frac{u^{2}}{2\\cdot\\:2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}}" }, { "type": "interim", "title": "$$2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}\\cdot\\:2=2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1}$$", "input": "2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}}\\cdot\\:2=\\:2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1}$$" ], "result": "=2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76daRfET2f6/Cw3ppZN1JknMiHdCfXsqtfrmk3UduuG4n9ypDQ6cTgHCnfTT6nWlpVdNK6b/PmQukzNdLVQkh7KxK5zL7DDt7dF16m8Uqihv+D0Mf+kipm0+oGxBmbA/0qn9dCGxGLPjnN+gRrSati/8//6/nV5O4fb8Xgwi7mapmTOlAjkdAmDWpSSp2DS3OcOrvHMmBBQJFpbYYrFqg4e9NG0mw+SpR/kFJvh+CE5V7/aMebnNjEJ4g1l9VybNYrw+ACfJwKzIMvonzdMkBT502JJUR8A94oCW6/UYwYT8=" } }, { "type": "step", "result": "=\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1}}" }, { "type": "interim", "title": "$$2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1}=2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}$$", "input": "2^{\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1:{\\quad}\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}$$", "input": "\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+1", "result": "=2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\ln\\left(2\\right)}{\\ln\\left(2\\right)}$$", "result": "=\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)}{\\ln\\left(2\\right)}+\\frac{1\\cdot\\:\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\ln\\left(u\\right)-\\ln\\left(2\\right)+1\\cdot\\:\\ln\\left(2\\right)}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Add similar elements: $$-\\ln\\left(2\\right)+1\\cdot\\:\\ln\\left(2\\right)=0$$", "result": "=\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76daRfET2f6/Cw3ppZN1JknMiHdCfXsqtfrmk3UduuG48pIstCRRTp9wZT3vHxmLodYPfXQvX4/bINBB8wSEQ0Vt1rIXxD4XxwCpXbo2OyOCBfYSP6D+1gqgZwfqGoCg5/z//r+dXk7h9vxeDCLuZqmZM6UCOR0CYNalJKnYNLc5w6u8cyYEFAkWlthisWqDhmSmvxU8zAvzF8nOUNQKOo1uV2F4mblu5IjU5fDMA1kCTXWz1DlkgSVk4l/BOy9+0" } }, { "type": "step", "result": "=\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mvSe5tp2adhB2rSdgRT7eXBB6tyk+QoeYEsxcIX2bTpjQsHC0SG4IWMmu1+n42vqkjyJ0dgskqXYx4DHhSkBEQGluW+ekU3B1Xg5xqZpjwh1g99dC9fj9sg0EHzBIRDRRsqkeES3SqCqjIoJM0Kr2w38WOtoVOPIzLjzCpM7kSaswjRs7Sz7H6LIiPOR61UyP8vQyhiD4JSfqjIvcQ7tihptmH1ReOkzVK2kyiYq0yuCBQN2cs1key3hk0/wPYgmDO5jKJEpApu3Vis/DpV2nQ9DYJA4IDKiq3NMrHU5vuAts1xK+/nNusxZ74S6C1gnFwE9SJQj74qo8RMrsqN++4F9hI/oP7WCqBnB+oagKDkWRUPDnIQXYMTDiaxDMpN7" } }, { "type": "interim", "title": "$$\\int\\:-\\frac{2}{u^{2}}\\cdot\\:\\frac{u^{2}}{2}du=\\int\\:-1du$$", "input": "\\int\\:-\\frac{2}{u^{2}}\\cdot\\:\\frac{u^{2}}{2}du", "steps": [ { "type": "interim", "title": "Multiply $$-\\frac{2}{u^{2}}\\cdot\\:\\frac{u^{2}}{2}\\::{\\quad}-1$$", "input": "-\\frac{2}{u^{2}}\\cdot\\:\\frac{u^{2}}{2}", "result": "=\\int\\:-1du", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{2u^{2}}{u^{2}\\cdot\\:2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{u^{2}}{u^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$u^{2}$$", "result": "=-1" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KJ367KHTHpp47MSSEdXrepClDm1NAdzZMwbM8/tu76MB0xn/umHY/dtWcdxZOmVsWb8uqNMy7ktYfP+jUy4AZv2i9gqKNBiEkMJvG7+cA4k5C2fUQubrlZRn1QHKx6IOU5JCCGZ98v0NjfXek/pS4c51+8sWjJh17lHzwFhaQVPM0wfl9zWSBOu7LIS5BessQajWQYMeFmH4Yn2byYHkUj8RdTP3Ys2P1kMDpUQHWTU=" } }, { "type": "step", "result": "=\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}}-\\int\\:-1du" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71HrZpDSxNlK+gk72r7bblqCBQN2cs1key3hk0/wPYgmBYbMcGQ9PXe9uLgUKO4irxwVaEBDGWkeJC1aIvqBRrQWo8nB41AsRPr2zXIEyUQlOWIWQi6ctAgJfOyKwdKH10/QYMzREewyYhmRoDar7ocjiRgY19cZKRIq7kidiAJiJMVebkv2FOLns9Ws5bPx8gcV/JiHMB28PazQGawgGp0xK1tyiWP3evVB8/1lyBCe3SmUvX4+ZxjvBdfdSGCsOzbAY5I2tDCJkXpH2GPYVtEB1mL+yfdsNlO7oMtLSCKv" } }, { "type": "step", "result": "=\\frac{1}{\\ln\\left(2\\right)}\\left(\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}}-\\int\\:-1du\\right)" }, { "type": "interim", "title": "$$\\int\\:-1du=-u$$", "input": "\\int\\:-1du", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=\\left(-1\\right)u" }, { "type": "step", "primary": "Simplify", "result": "=-u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{\\ln\\left(2\\right)}\\left(\\frac{u^{2}}{2^{\\frac{\\ln\\left(u\\right)}{\\ln\\left(2\\right)}}}-\\left(-u\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=2^{\\sqrt{x}}$$", "result": "=\\frac{1}{\\ln\\left(2\\right)}\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}-\\left(-2^{\\sqrt{x}}\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\ln\\left(2\\right)}\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}-\\left(-2^{\\sqrt{x}}\\right)\\right):{\\quad}\\frac{2\\cdot\\:2^{\\sqrt{x}}}{\\ln\\left(2\\right)}$$", "input": "\\frac{1}{\\ln\\left(2\\right)}\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}-\\left(-2^{\\sqrt{x}}\\right)\\right)", "result": "=\\frac{2\\cdot\\:2^{\\sqrt{x}}}{\\ln\\left(2\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1}{\\ln\\left(2\\right)}\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)=\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}$$", "input": "1\\cdot\\:\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)=\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)$$", "result": "=\\left(\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/j5hVgER6WFXHVJziYJml1Cizn1KDb2P8BdRJ2wYHET1n+iaCQf21BtxLktGELkcdlmKdbt6K12/iwb4CKHa+JAEEMq9UA5X2WGO4uBmwiKnjvSiRouMKLeLqgMnHINXLTrWWMFI8l4Q07DZ5+hJa0hqcfXty6JYd1pWIgQ3Zjj5clj4M2/OTxFENaAeEqrBH7WhG7LWeACV8fPZWx9iiyWD5WhL5GiTpDkwbHgKtnfsbEkKdOwL8IJ6hf9BFhg175crGIZFDw8+IITVIowW4YxRfhq4bXv9lKhZM7D7OKc8tZppPtQdWwBePX1z3MkIy4ecvvuQZhcVG929nJgSrtnlE25ss8QXZ+tAsEeaHzDPQYr85KxVZmROw1ynEPf/W5fcPDIZEBTPaHceNmJ92wOqWO2czV7V2gVuPSkaod+n1BvU6/gGOW5wMh2wCV3OAVWzK5yiaiaO+9pNkIrqcwq1ieCYNjhcPIbM7bNqgpzkE7Czs7bZnm+4hG9ehXHD8hflhAjx+fXUP+q/IrXcjrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}+2^{\\sqrt{x}}}{\\ln\\left(2\\right)}" }, { "type": "interim", "title": "$$\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}=2^{\\sqrt{x}}$$", "input": "\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}}", "steps": [ { "type": "interim", "title": "$$2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}=2^{\\sqrt{x}}$$", "input": "2^{\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}}", "steps": [ { "type": "interim", "title": "$$\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}=\\sqrt{x}$$", "input": "\\frac{\\ln\\left(2^{\\sqrt{x}}\\right)}{\\ln\\left(2\\right)}", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(2^{\\sqrt{x}}\\right):{\\quad}\\ln\\left(2\\right)\\sqrt{x}$$", "input": "\\ln\\left(2^{\\sqrt{x}}\\right)", "result": "=\\frac{\\ln\\left(2\\right)\\sqrt{x}}{\\ln\\left(2\\right)}", "steps": [ { "type": "step", "primary": "Apply log rule $$\\log_{a}\\left(x^b\\right)=b\\cdot\\log_{a}\\left(x\\right),\\:\\quad$$ assuming $$x\\:\\geq\\:0$$", "result": "=\\ln\\left(2\\right)\\sqrt{x}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$\\ln\\left(2\\right)$$", "result": "=\\sqrt{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VUF3jOsvvk/OM0GietP3oM1mXGUpwzAt/3k75V/MNe9dhNGpYuc0It3Km1u+fq1nq47vuWedXv2WUg94ER8IwfUX2UFTxNde5SO8y5RAo5AHk1eLttM1tdVeTyzsC+lrCrWJ4Jg2OFw8hszts2qCnOQTsLOzttmeb7iEb16FccMA19Oca6fdkRRUFxahS7QN" } }, { "type": "step", "result": "=2^{\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76daRfET2f6/Cw3ppZN1JkqSzYqNFMGXHjc0UZEcryPpK4W2EfWhalDlINWnYUMvkzRqDxPUzBN6vjj5oJL9kUJCMbLzygEIofDPXUdA6yNROjPXSg46oNj8sFa+h7LaVHYrRD0C29wFG6+jIGKWbLs9BivzkrFVmZE7DXKcQ9/84w7UPOH7FCG4+HmKorQIT7cjcR2yMRai0wVGzWKJfIA==" } }, { "type": "step", "result": "=\\frac{\\left(2^{\\sqrt{x}}\\right)^{2}}{2^{\\sqrt{x}}}" }, { "type": "interim", "title": "$$\\left(2^{\\sqrt{x}}\\right)^{2}=2^{2\\sqrt{x}}$$", "input": "\\left(2^{\\sqrt{x}}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=2^{\\sqrt{x}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s707ufHgYIitXoW6J0GmVbtuiEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPITMP/ZRrzMKFv41n8QLmoDF8D13SVCUc5ZpUzmAv5QAEjNy10cfkZs0B8PNOlTUSEO/qpKRMkDiLTx8PkeV+ehze+2eTdER4kDISFpALfr2w=" } }, { "type": "step", "result": "=\\frac{2^{2\\sqrt{x}}}{2^{\\sqrt{x}}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}\\:=\\:x^{a-b}$$", "secondary": [ "$$\\frac{2^{2\\sqrt{x}}}{2^{\\sqrt{x}}}=2^{2\\sqrt{x}-\\sqrt{x}}=2^{\\sqrt{x}}$$" ], "result": "=2^{\\sqrt{x}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7koEVo7NSH4GzEEqnv4shrsZWx3EUllWXkri0CPcDQ1n58FfaEs82q2QxRjuMld6wYhig9JVLH5AMZ/omdWfkXEP9m/C5xOb7vTC2AAC+Pv8JQJZuTAY5js+oqjdT8kslw2GccSGvKmkfA5R6EKQRrf8//6/nV5O4fb8Xgwi7maoUKXO+7nhjXTtO9DlesMTfNrFzOuR3U0/yBHR08rKv0h20oA7goJCrpj7zxaMqwAEzthBHFpKE/CdbH+QOYj+Patxeia/MdHiKsAwiJGRPA2NqRlxBl/P5P/lepu5L+6w=" } }, { "type": "step", "result": "=\\frac{2^{\\sqrt{x}}+2^{\\sqrt{x}}}{\\ln\\left(2\\right)}" }, { "type": "step", "primary": "Add similar elements: $$2^{\\sqrt{x}}+2^{\\sqrt{x}}=2\\cdot\\:2^{\\sqrt{x}}$$", "result": "=\\frac{2\\cdot\\:2^{\\sqrt{x}}}{\\ln\\left(2\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ziQ6xHlyk6f0M0x/f9gYtjhJo7llyazSGfz1gelnAVQpgFP25n7EcP/lvEdCpcWJF2s7NOWqMCd3hoNleH75CinAinYk8Q/GGherXT0pahDbUkq1l56EWZLcZdiUEzB/OtmWaXRo6NHJKHgzA80nzyAn9lkDfZkicUGkO3EF+IpdgYKaIEX32V6yZdn2Ih8nga3sCMcmkk8sYQB82VkY4jDdHo+wnkNbP3FSyGClQdNkS3dlcCKpQTQcheuut7MkdkqXaZXgCXO2IClMAWKDPU22+beuEy5yqQ1MZB1mcFc4SaO5Zcms0hn89YHpZwFUKYBT9uZ+xHD/5bxHQqXFiRdrOzTlqjAnd4aDZXh++QopwIp2JPEPxhoXq109KWoQ21JKtZeehFmS3GXYlBMwfzgESV9orXMmZVzo1QOK9mU=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{2\\cdot\\:2^{\\sqrt{x}}}{\\ln\\left(2\\right)}+C", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "practiceLink": "/practice/integration-practice#area=main&subtopic=Integration%20By%20Parts", "practiceTopic": "Integration by Parts" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{2\\cdot 2^{\\sqrt{x}}}{\\ln(2)}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }