integral from 1 to 2 of xsqrt(x^2-1)

{ "query": { "display": "$$\\int_{1}^{2}x\\sqrt{x^{2}-1}dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{1}^{2}x\\sqrt{x^{2}-1}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\sqrt{3}", "decimal": "1.73205…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{1}^{2}x\\sqrt{x^{2}-1}dx=\\sqrt{3}$$", "input": "\\int_{1}^{2}x\\sqrt{x^{2}-1}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int_{1}^{2}x\\sqrt{x^{2}-1}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: $$u=\\sqrt{x^{2}-1}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\frac{x}{\\sqrt{x^{2}-1}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x^{2}-1}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{x^{2}-1}}\\frac{d}{dx}\\left(x^{2}-1\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x^{2}-1}\\right)", "result": "=\\frac{1}{2\\sqrt{x^{2}-1}}\\frac{d}{dx}\\left(x^{2}-1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=x^{2}-1$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(x^{2}-1\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sqrt{u}\\right)=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\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": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{1}{2}u^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{u}}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{1}{2}-1}=u^{-\\frac{1}{2}}$$", "input": "u^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{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\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{u}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{u}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2rvsVWktCxa7XSYzIK90x3+aTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{u}}\\frac{d}{dx}\\left(x^{2}-1\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}-1$$", "result": "=\\frac{1}{2\\sqrt{x^{2}-1}}\\frac{d}{dx}\\left(x^{2}-1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjasHTJ4Bo7yMP4qgzIlGEQz+MOmdrBX/prCtlLheMuaZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJXv++bSprT8DRLjDQza+XRVtP4dW/8YT8X7GJ2F6L9zLzaOOlj9goo8V013IVnrkLL1YJ/Ns/Q/mFf1axOuM43jRSpN33oxZMojoqvYhvSJAFuBNke0eZANmQMdPqVsU1Pq9PUUBMmxzxm22kIDr81B" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}-1\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)-\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=2x-0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{x^{2}-1}}\\cdot\\:2x" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{x^{2}-1}}\\cdot\\:2x:{\\quad}\\frac{x}{\\sqrt{x^{2}-1}}$$", "input": "\\frac{1}{2\\sqrt{x^{2}-1}}\\cdot\\:2x", "result": "=\\frac{x}{\\sqrt{x^{2}-1}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{2\\sqrt{x^{2}-1}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1\\cdot\\:x}{\\sqrt{x^{2}-1}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=\\frac{x}{\\sqrt{x^{2}-1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13CxPCgwn69xsaEYCj3/USRH5tOYrMhCJMBBPrwsYfxOuCq47vuWedXv2WUg94ER8IwYQO7+z45GqFJXp4sUC1pO3Bh9ZBIdXKMmXIkyyLRwqeRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oHIcXcImz+eH9oVkJvPejdx1GA8Hrnakmsj7FyJ+lyZKzTLYCkr3cyMfwv5DlY8ks=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{x}{\\sqrt{x^{2}-1}}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{\\sqrt{x^{2}-1}}{x}du$$" }, { "type": "step", "result": "=\\int\\:xu\\frac{\\sqrt{x^{2}-1}}{x}du" }, { "type": "step", "primary": "$$u=\\sqrt{x^{2}-1}$$", "result": "=\\int\\:xu\\frac{u}{x}du" }, { "type": "interim", "title": "Simplify $$xu\\frac{u}{x}:{\\quad}u^{2}$$", "input": "xu\\frac{u}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{uxu}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=uu" }, { "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": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:u^{2}du" }, { "type": "step", "primary": "Adjust integral boundaries:" }, { "type": "interim", "title": "$$x=1\\quad\\Rightarrow\\:u=0$$", "input": "u=\\sqrt{x^{2}-1}", "steps": [ { "type": "step", "primary": "Plug in $$x=1$$", "result": "=\\sqrt{1^{2}-1}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-1}" }, { "type": "step", "primary": "Subtract the numbers: $$1-1=0$$", "result": "=\\sqrt{0}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{0}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GYHXbXcKbupVyqVMPpRTyaiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhFiaOLVxaerb4I/zP7NCc4ZefYGPUkli9MVd3HFvHxPN" } }, { "type": "interim", "title": "$$x=2\\quad\\Rightarrow\\:u=\\sqrt{3}$$", "input": "u=\\sqrt{x^{2}-1}", "steps": [ { "type": "step", "primary": "Plug in $$x=2$$", "result": "=\\sqrt{2^{2}-1}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4-1}" }, { "type": "step", "primary": "Subtract the numbers: $$4-1=3$$", "result": "=\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GYHXbXcKbupVyqVMPpRTyaiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrFBc9Nmoth9+skvxKsVPOujbHnKjB8Fg2cHxK2Gh4mQVcCpsjPix7bEqEy5QF+6nmumUfTGumNyut34etZhmk/w==" } }, { "type": "step", "result": "=\\int_{0}^{\\sqrt{3}}u^{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73gNLKLOs8VbbE8cGl6PJRz7JnMwrouhNU5V34Ng/mouB42dCZngAt2UhJjMVkTWLsN5vRWJvr0V/8T6RtR0CSAgHP4YqWkh5OOHaSrQ/l8R41cMteKH5QlDE/dUD+SpBiBVds/xqoiCqRZh214pM0KBBTEk/JQ2cZ9WKuRzClU7Bi3jMVZeecTGMRH4ygWrbpXEFJTDoZKkLxfTvEPP2eY=" } }, { "type": "step", "result": "=\\int_{0}^{\\sqrt{3}}u^{2}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{\\sqrt{3}}u^{2}du", "result": "=[\\frac{u^{3}}{3}]_{0}^{\\sqrt{3}}", "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^{2+1}}{2+1}]_{0}^{\\sqrt{3}}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{2+1}}{2+1}:{\\quad}\\frac{u^{3}}{3}$$", "input": "\\frac{u^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=[\\frac{u^{3}}{3}]_{0}^{\\sqrt{3}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zXEm1MbHnl5iZXlJ+3KXls7TJ5CaiVLOEKdqLIvqKmfqPySOWwYKYBMTeddEwOXMqHakgmJ1D/iojSuxNs5WIjAqOvY2yb2fmFq+DOa/EdTXoyzO9FC1NTbfyRK2KqK4mtC1ggrpw4qEhYuVtggMeBFKk3fejFkyiOiq9iG9IkAW4E2R7R5kA2ZAx0+pWxTUzmaCTF+08KtdZkgx0pZ/mA=" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\sqrt{3}$$", "input": "[\\frac{u^{3}}{3}]_{0}^{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "interim", "title": "$$\\lim_{u\\to\\:0+}\\left(\\frac{u^{3}}{3}\\right)=0$$", "input": "\\lim_{u\\to\\:0+}\\left(\\frac{u^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=0$$", "result": "=\\frac{0^{3}}{3}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{0^{3}}{3}:{\\quad}0$$", "input": "\\frac{0^{3}}{3}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{3}=0$$" ], "result": "=\\frac{0}{3}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qBRpSLsqhPH9RPF5elXY++T0JTwwJam8DFLG9sjVKUME5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXRjRfJ/eTycHczxN2mOTuFsEk7YwRD0hobDQfAqSn6vxw==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:\\sqrt{3}-}\\left(\\frac{u^{3}}{3}\\right)=\\sqrt{3}$$", "input": "\\lim_{u\\to\\:\\sqrt{3}-}\\left(\\frac{u^{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=\\sqrt{3}$$", "result": "=\\frac{\\left(\\sqrt{3}\\right)^{3}}{3}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{\\left(\\sqrt{3}\\right)^{3}}{3}:{\\quad}\\sqrt{3}$$", "input": "\\frac{\\left(\\sqrt{3}\\right)^{3}}{3}", "result": "=\\sqrt{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(\\sqrt{3}\\right)^{3}:{\\quad}3\\sqrt{3}$$", "input": "\\left(\\sqrt{3}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:3=\\frac{3}{2}$$", "input": "\\frac{1}{2}\\cdot\\:3", "result": "=3^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8SPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBnpkZpWG0liIE8buj+sqXg2RLd2VwIqlBNByF6663sySF76Eydb/wnfqEDL8zJUMWNJdlTvBkWjQ+TKBgKD6qgLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$3^{\\frac{3}{2}}=3\\sqrt{3}$$", "input": "3^{\\frac{3}{2}}", "result": "=3\\sqrt{3}", "steps": [ { "type": "step", "primary": "$$3^{\\frac{3}{2}}=3^{1+\\frac{1}{2}}$$", "result": "=3^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=3^{1}\\cdot\\:3^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=3\\sqrt{3}" } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7x/Y49pORHlTzA5k3OgZjWHyRHuGw7+tM5METTDj6vVHFAQO2o5E4TQsyBiKgckoThUulnmFIQQyhpb8h/WI6dHql8XXPq6bNQlMm+36iNhnp8crKNPHHzSD/J1PULp1PRi/VnLHgUPxWDS2joBFKUQ==" } }, { "type": "step", "result": "=\\frac{3\\sqrt{3}}{3}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kLn7iC+AOhHStEx/knbw2XNJtd3Uno24tNB2ZsAaIMIgJ/ZZA32ZInFBpDtxBfiKSMge4f0/c1Ud4uYOHSaPOxSYeD6JDEaUbEpYNvzKYnMeKYFFhGodJZ4mS5Q+F9MLEM4Gz9d2Tc0aOjjwAZmM7ADxQlycBkVUnG8h1qWBr3q/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\sqrt{3}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\sqrt{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73gNLKLOs8VbbE8cGl6PJRz7JnMwrouhNU5V34Ng/mouWO6LWPo1e6CFlWFbliAlcVqmQTbqct5X4+XES7R+yKKOY/JMhQU6i+Hzirqc60lKCL5eJNXi2Y/QfPoBrCDTvnql8XXPq6bNQlMm+36iNhmIsAHIkXRe6v0CQ+qFmJtW3T8KO0JtVOqEEoDvY/aXow==" } }, { "type": "step", "result": "=\\sqrt{3}" } ], "meta": { "solvingClass": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }