integral of (sqrt(4-x^2))/x

{ "query": { "display": "$$\\int\\:\\frac{\\sqrt{4-x^{2}}}{x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{\\sqrt{4-x^{2}}}{x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{\\sqrt{4-x^{2}}}{x}dx=-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{2}}+C$$", "input": "\\int\\:\\frac{\\sqrt{4-x^{2}}}{x}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\sqrt{4-x^{2}}}{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: $$u=\\sqrt{4-x^{2}}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-\\frac{x}{\\sqrt{4-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{4-x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{4-x^{2}}}\\frac{d}{dx}\\left(4-x^{2}\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{4-x^{2}}\\right)", "result": "=\\frac{1}{2\\sqrt{4-x^{2}}}\\frac{d}{dx}\\left(4-x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=4-x^{2}$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(4-x^{2}\\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(4-x^{2}\\right)" }, { "type": "step", "primary": "Substitute back $$u=4-x^{2}$$", "result": "=\\frac{1}{2\\sqrt{4-x^{2}}}\\frac{d}{dx}\\left(4-x^{2}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqws7SQcG//90aDq2y0nVyst9MMddrkFuNTLaPWMeYe3Z3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJXv++bSprT8DRLjDQza+XRVK8bX5dVPv1mw7bndgYyo4aKwxODUNSw+sWQrjsE7uIE5Nrp7QlQvtOoqamnUHSv5RSpN33oxZMojoqvYhvSJAFuBNke0eZANmQMdPqVsU1Pq9PUUBMmxzxm22kIDr81B" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4-x^{2}\\right)=-2x$$", "input": "\\frac{d}{dx}\\left(4-x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(4\\right)-\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\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+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "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": "step", "result": "=0-2x" }, { "type": "step", "primary": "Simplify", "result": "=-2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{4-x^{2}}}\\left(-2x\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{4-x^{2}}}\\left(-2x\\right):{\\quad}-\\frac{x}{\\sqrt{4-x^{2}}}$$", "input": "\\frac{1}{2\\sqrt{4-x^{2}}}\\left(-2x\\right)", "result": "=-\\frac{x}{\\sqrt{4-x^{2}}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2\\sqrt{4-x^{2}}}\\cdot\\:2x" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:2x}{2\\sqrt{4-x^{2}}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1\\cdot\\:x}{\\sqrt{4-x^{2}}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=-\\frac{x}{\\sqrt{-x^{2}+4}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13C05eUZ+miyV2wo6fK1k/Zk6O6Dd3zpuSnKXNDXc0BYg4ebTKSUwQXHFFpEsrzHtUqUlp37JZxv9+CIGyLDYI+uDk5PyMmrrndFr6EuaTQELKgQUxJPyUNnGfVirkcwpVO0BvnUFx4T5rmmmJUxss5NmDbMfNMThqBPZI/ssF14xUWF/7SCsO18QgBd23evdIfSS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-\\frac{x}{\\sqrt{4-x^{2}}}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-\\frac{\\sqrt{4-x^{2}}}{x}\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{u}{x}\\left(-\\frac{\\sqrt{4-x^{2}}}{x}\\right)du" }, { "type": "step", "primary": "$$u=\\sqrt{4-x^{2}}$$", "result": "=\\int\\:\\frac{u}{x}\\left(-\\frac{u}{x}\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{u}{x}\\left(-\\frac{u}{x}\\right):{\\quad}-\\frac{u^{2}}{x^{2}}$$", "input": "\\frac{u}{x}\\left(-\\frac{u}{x}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{u}{x}\\cdot\\:\\frac{u}{x}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{uu}{xx}" }, { "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{u^{2}}{xx}" }, { "type": "interim", "title": "$$xx=x^{2}$$", "input": "xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74tglIeOAQqOR+OcPiyzQUczBWJotReR4P4m6RE6FZ2M7Aq6fHyeqJtW5OKbXVcT+IBF/biSmVq3Z2pV/8nBrAiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=-\\frac{u^{2}}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-\\frac{u^{2}}{x^{2}}du" }, { "type": "interim", "title": "$$u=\\sqrt{4-x^{2}}\\quad\\Rightarrow\\quad\\:x^{2}=-u^{2}+4$$", "input": "\\sqrt{4-x^{2}}=u", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}4-x^{2}=u^{2}$$", "input": "\\sqrt{4-x^{2}}=u", "result": "4-x^{2}=u^{2}", "steps": [ { "type": "step", "result": "\\left(\\sqrt{4-x^{2}}\\right)^{2}=u^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{4-x^{2}}\\right)^{2}:{\\quad}4-x^{2}$$", "input": "\\left(\\sqrt{4-x^{2}}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(4-x^{2}\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(4-x^{2}\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=4-x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAia4Jdx/cbn+LiAS76SGE7OgO2nxgRIn8nSu8AsLJaIIvReWy3ZWghbIQ97+Ky8uxHBVNKmeFzMIUpZt5JxZIw1De1A9ekahUIWtvNWrN4v94XgPrrkCq8jPZd2BMRR8qv4TMpnSw4KVOX2qrGO8xpzc=" } }, { "type": "step", "result": "4-x^{2}=u^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWN+JAs0h0ZUpukxIbATVWZ3XDg1Z3KZYaDri4vsnz806EwwsKFQxG7qoQVnFpC7BA5IjE0IfGwFn4DjmRHa83ek3kCh3oevUunZ7/b0qFKBQtt47Cfbjwk5Ca+gEVzKSfvUUGvlRczHDvLjdst2blpA==" } }, { "type": "interim", "title": "Solve $$4-x^{2}=u^{2}:{\\quad}x^{2}=-u^{2}+4$$", "input": "4-x^{2}=u^{2}", "steps": [ { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "4-x^{2}=u^{2}", "result": "-x^{2}=u^{2}-4", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "4-x^{2}-4=u^{2}-4" }, { "type": "step", "primary": "Simplify", "result": "-x^{2}=u^{2}-4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-x^{2}=u^{2}-4", "result": "x^{2}=-u^{2}+4", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-x^{2}}{-1}=\\frac{u^{2}}{-1}-\\frac{4}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-x^{2}}{-1}=\\frac{u^{2}}{-1}-\\frac{4}{-1}", "result": "x^{2}=-u^{2}+4", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-x^{2}}{-1}:{\\quad}x^{2}$$", "input": "\\frac{-x^{2}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{x^{2}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ONL1vVgt9HKkg+QCCD7OjDFSpzwRDPIFcbKdvhSPhgvMwViaLUXkeD+JukROhWdjOwKunx8nqibVuTim11XE/h429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCV5VB7agcBAJG49JU4/CHLw==" } }, { "type": "interim", "title": "Simplify $$\\frac{u^{2}}{-1}-\\frac{4}{-1}:{\\quad}-u^{2}+4$$", "input": "\\frac{u^{2}}{-1}-\\frac{4}{-1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{u^{2}-4}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{2}-4}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-\\left(u^{2}-4\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-u^{2}-\\left(-4\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-u^{2}+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77Di5zkSxEnEr/M7OgI79vbWKPvAVA5Mdn5jekq4gFrp8kR7hsO/rTOTBE0w4+r1Rx2TqcRquPGhcQqCcnrizmQQQiwsX+hz3oFr5iYFwdLoeNvb7k0sVmuwf19w9aD9NwFkJl/hQGUOe3kPB8GwiQnWbpvwCMKvmolUp3trufUMCCcfIqNfKdjVG9Bf4qTSD" } }, { "type": "step", "result": "x^{2}=-u^{2}+4" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x^{2}=-u^{2}+4" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}x^{2}=-u^{2}+4\\:\\left\\{u\\ge\\:0\\right\\}$$", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\sqrt{4-x^{2}}=u$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug$${\\quad}x^{2}=-u^{2}+4:{\\quad}\\sqrt{4-\\left(-u^{2}+4\\right)}=u{\\quad}\\Rightarrow{\\quad}u\\ge\\:0$$", "input": "\\sqrt{4-\\left(-u^{2}+4\\right)}=u", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}u^{2}=u^{2}$$", "input": "\\sqrt{4-\\left(-u^{2}+4\\right)}=u", "result": "u^{2}=u^{2}", "steps": [ { "type": "step", "result": "\\left(\\sqrt{4-\\left(-u^{2}+4\\right)}\\right)^{2}=u^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{4-\\left(-u^{2}+4\\right)}\\right)^{2}:{\\quad}u^{2}$$", "input": "\\left(\\sqrt{4-\\left(-u^{2}+4\\right)}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(4-\\left(-u^{2}+4\\right)\\right)^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(4-\\left(-u^{2}+4\\right)\\right)^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=4-\\left(-u^{2}+4\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": 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"$$4-4=0$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7w9qear4seNTTD1F+IKU/h15WboOlMxxEYQ6NYGhPEBRvHbq4gNPYkkADD88xoNcLEPtnTtTi5i6FVgwrEZkiMvC30sSftAIFS6Qkpy19IkquoEGBFRdPJ6d5qlzxxz6NvmDusxhkeu63Kwnyy8XqBA==" } }, { "type": "step", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAicNQf3fGWOrRt3SJwUJRgouq/y37z44Ci/wB3YLNRUeKdQxShwOg3W+URNAMu5sPa6VYwrogLl29RT6HYd2NJ316pfF1z6umzUJTJvt+ojYZjmLRtfamxNlPDopLD4I+m2a2xDkZCIkY9cZlDGv7ygf6LRTHeWmu9fnQIV+XmusB" } }, { "type": "step", "result": "u^{2}=u^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QW5M97a5PzrRjkplhkc40MG7rCv9q/5eQKSbi4CewZ5tv6hniYomRHCtYYrUM7LOYsq7fhGHm4HbOqytP79nuBhu9sGZu5A1MXROmEpnxG69oR2lFfMTfVgjZtzh42dsFQOf26sr1IOehEBVbNXgdV4g==" } }, { "type": "step", "primary": "Both sides are equal", "result": "\\mathrm{True\\:for\\:all}\\:u" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}u<0\\:$$False$$,\\:\\:u=0\\:$$True$$,\\:\\:u>0\\:$$True", "input": "\\sqrt{4-\\left(-u^{2}+4\\right)}=u", "steps": [ { "type": "step", "primary": "Combine domain interval with solution interval:", "result": "\\mathrm{True\\:for\\:all}\\:u" }, { "type": "interim", "title": "Find the function intervals:$${\\quad}u<0,\\:u=0,\\:u>0$$", "input": "\\sqrt{4-\\left(-u^{2}+4\\right)}=u", "steps": [ { "type": "step", "primary": "Find the even roots arguments zeroes:" }, { "type": "interim", "title": "Solve $$4-\\left(-u^{2}+4\\right)=0:{\\quad}u=0$$", "input": "4-\\left(-u^{2}+4\\right)=0", "steps": [ { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "4-\\left(-u^{2}+4\\right)=0", "result": "-\\left(-u^{2}+4\\right)=-4", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "4-\\left(-u^{2}+4\\right)-4=0-4" }, { "type": "step", "primary": "Simplify", "result": "-\\left(-u^{2}+4\\right)=-4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-\\left(-u^{2}+4\\right)=-4", "result": "-u^{2}+4=4", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-\\left(-u^{2}+4\\right)}{-1}=\\frac{-4}{-1}" }, { "type": "step", "primary": "Simplify", "result": "-u^{2}+4=4" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "-u^{2}+4=4", "result": "-u^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "-u^{2}+4-4=4-4" }, { "type": "step", "primary": "Simplify", "result": "-u^{2}=0" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-u^{2}=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-u^{2}=0", "result": "u^{2}=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-u^{2}}{-1}=\\frac{0}{-1}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq" } }, { "type": "step", "result": "u=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=0" }, { "type": "step", "primary": "The intervals are defined around the zeroes:", "result": "u<0,\\:u=0,\\:u>0" }, { "type": "step", "primary": "Combine intervals with domain", "result": "u<0,\\:u=0,\\:u>0" } ], "meta": { "interimType": "Abs Find Intervals Title 0Eq" } }, { "type": "step", "primary": "Check the solutions by plugging them into $$\\sqrt{4-\\left(-u^{2}+4\\right)}=u$$<br/>Remove the ones that don't agree with the equation.", "secondary": [ "Plug$${\\quad}u<0:{\\quad}\\sqrt{4-\\left(-u^{2}+4\\right)}\\ne\\:u{\\quad}\\Rightarrow{\\quad}$$False", "Plug$${\\quad}u=0:{\\quad}\\sqrt{4-\\left(-u^{2}+4\\right)}=u{\\quad}\\Rightarrow{\\quad}$$True", "Plug$${\\quad}u>0:{\\quad}\\sqrt{4-\\left(-u^{2}+4\\right)}=u{\\quad}\\Rightarrow{\\quad}$$True" ] } ], "meta": { "interimType": "Verifying Solutions Title 0Eq" } }, { "type": "step", "primary": "The solution is", "result": "u\\ge\\:0" } ], "meta": { "solvingClass": "Equations", "interimType": "Check One Solution Specific 3Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "primary": "The solution is", "result": "x^{2}=-u^{2}+4\\:\\left\\{u\\ge\\:0\\right\\}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:-\\frac{u^{2}}{-u^{2}+4}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7++Ya7Ho8qtLONozuWT48l+gZ75iObD1O7bF2LOZcpboK576WENGsoCX8pronHaqWSlQj8vi8h0NbzZABdtckVuxnvL7f2wjac+8KPowiAvKtUGIZi9ebLbpDEqv7LukVyFaaxlZFO2ParRiOJfKIW12dOCV9LmnoleDPKqfMJM2fQ5Q/JdC0UaI6OYSv9c0zsajPLMT6+iiCYQ7FcCsRHskt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\int\\:-\\frac{u^{2}}{-u^{2}+4}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": "=-\\int\\:\\frac{u^{2}}{-u^{2}+4}du" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{u^{2}}{-u^{2}+4}du", "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: $$u=2v$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=4,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=2\\quad\\Rightarrow\\quad$$substitute $$x=2u$$" }, { "type": "interim", "title": "$$\\frac{du}{dv}=2$$", "input": "\\frac{d}{dv}\\left(2v\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dv}{dv}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqzgBS7FKezVF3apKLAK1r3ZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51glGyPClOpED27XC+7kjsHQ" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{\\left(2v\\right)^{2}}{-\\left(2v\\right)^{2}+4}\\cdot\\:2dv" }, { "type": "interim", "title": "Simplify $$\\frac{\\left(2v\\right)^{2}}{-\\left(2v\\right)^{2}+4}\\cdot\\:2:{\\quad}\\frac{2v^{2}}{-v^{2}+1}$$", "input": "\\frac{\\left(2v\\right)^{2}}{-\\left(2v\\right)^{2}+4}\\cdot\\:2", "steps": [ { "type": "interim", "title": "$$\\frac{\\left(2v\\right)^{2}}{-\\left(2v\\right)^{2}+4}=\\frac{v^{2}}{-v^{2}+1}$$", "input": "\\frac{\\left(2v\\right)^{2}}{-\\left(2v\\right)^{2}+4}", "steps": [ { "type": "interim", "title": "$$\\left(2v\\right)^{2}=4v^{2}$$", "input": "\\left(2v\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}v^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZLojdXH40Ct3TDisYH6Toc0ag8T1MwTer44+aCS/ZFBTTJO+BjoXkYCd+W4dWsWmxyWsZgw1kzIMlALnznRBpDCxcwp/0Qhjtf2O5xeKzTc=" } }, { "type": "step", "result": "=\\frac{\\left(2v\\right)^{2}}{-4v^{2}+4}" }, { "type": "interim", "title": "$$\\left(2v\\right)^{2}=2^{2}v^{2}$$", "input": "\\left(2v\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}v^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZLojdXH40Ct3TDisYH6Toc0ag8T1MwTer44+aCS/ZFCr/ewY9m7Q0rBdUG9bYbGz/z//r+dXk7h9vxeDCLuZqma8hmhbZ3WHG9Sh5tM0CMTF3DenEWojLSGXYMDcAl7b" } }, { "type": "step", "result": "=\\frac{2^{2}v^{2}}{-4v^{2}+4}" }, { "type": "interim", "title": "Factor $$-4v^{2}+4:{\\quad}4\\left(-v^{2}+1\\right)$$", "input": "-4v^{2}+4", "result": "=\\frac{2^{2}v^{2}}{4\\left(-v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-4v^{2}+4\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$4$$", "result": "=4\\left(-v^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}v^{2}}{2^{2}\\left(-v^{2}+1\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=\\frac{v^{2}}{-v^{2}+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71bb3b2vXkUKbj+/HudDKk/P7a/NxOiHtZo6d3P6nPYYtOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOEmzHBMc5HkJqHtuVKSyfpS747WRaboUHQDi6e+m4VRaR4uSf8YaTqfndPKAGPEqgSoh9I4OkRrqypIlgEoMuX9fts+o2hyMsrPEPK8CMQ2OAECsdyFgm1SbPBiHkMZNSQ==" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{v^{2}}{-v^{2}+1}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{v^{2}\\cdot\\:2}{-v^{2}+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{2v^{2}}{-v^{2}+1}dv" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=-\\int\\:\\frac{2v^{2}}{-v^{2}+1}dv" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-2\\cdot\\:\\int\\:\\frac{v^{2}}{-v^{2}+1}dv" }, { "type": "interim", "title": "$$\\frac{v^{2}}{-v^{2}+1}=\\frac{1}{-v^{2}+1}-1$$", "input": "\\frac{v^{2}}{-v^{2}+1}", "steps": [ { "type": "step", "primary": "$$\\frac{v^{2}}{-v^{2}+1}=\\frac{v^{2}+\\left(-v^{2}+1\\right)}{-v^{2}+1}-1$$", "result": "=\\frac{v^{2}+\\left(-v^{2}+1\\right)}{-v^{2}+1}-1", "meta": { "title": { "extension": "Apply the following algebraic property$$:{\\quad}\\frac{a}{1-a}=\\frac{1}{1-a}-1$$<br/>$$\\frac{a}{1-a}=\\frac{1-1+a}{1-a}=\\frac{1}{1-a}+\\frac{-1+a}{1-a}=\\frac{1}{1-a}+\\frac{-\\left(1-a\\right)}{1-a}=\\frac{1}{1-a}-1$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{v^{2}+\\left(-v^{2}+1\\right)}{-v^{2}+1}-1:{\\quad}\\frac{1}{-v^{2}+1}-1$$", "input": "\\frac{v^{2}+\\left(-v^{2}+1\\right)}{-v^{2}+1}-1", "result": "=\\frac{1}{-v^{2}+1}-1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{v^{2}-v^{2}+1}{-v^{2}+1}-1" }, { "type": "step", "primary": "Add similar elements: $$v^{2}-v^{2}=0$$", "result": "=\\frac{1}{-v^{2}+1}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=-2\\cdot\\:\\int\\:\\frac{1}{-v^{2}+1}-1dv" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=-2\\left(\\int\\:\\frac{1}{-v^{2}+1}dv-\\int\\:1dv\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{-v^{2}+1}dv=\\frac{\\ln\\left|v+1\\right|}{2}-\\frac{\\ln\\left|v-1\\right|}{2}$$", "input": "\\int\\:\\frac{1}{-v^{2}+1}dv", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{-v^{2}+1}dv=\\frac{\\ln\\left|v+1\\right|}{2}-\\frac{\\ln\\left|v-1\\right|}{2}$$", "result": "=\\frac{\\ln\\left|v+1\\right|}{2}-\\frac{\\ln\\left|v-1\\right|}{2}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74Udxi4BZ1D2UC75vdfF8R4oqNoThdbh+jtSm4TJuabMpNyF73Y5cvCu+OrF4zWlvxZRiS8ynZP43c+X2Ms4DJdakTZfFWzXokgxn5FqCM8f+eAOtSE4DE0150N/XWNp65vK1nZPnNmaOfkjs4004lb0aZXhd2az3jrVrxOo81+do3oe/oyhMy2+1TQhDBd2f1ea3JYPJXGh8ZeMXXndkJdA6RidAWm/5usEvaBpv3VD9T+d8izgLieHuG3s66Vpc6yOoNq987B36TDpegK3OcnrNOVKgWC0qpnjVyTOQ3+w8CiV4ORVoCQuLN8W5NWVPg==" } }, { "type": "interim", "title": "$$\\int\\:1dv=v$$", "input": "\\int\\:1dv", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:v" }, { "type": "step", "primary": "Simplify", "result": "=v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-2\\left(\\frac{\\ln\\left|v+1\\right|}{2}-\\frac{\\ln\\left|v-1\\right|}{2}-v\\right)" }, { "type": "interim", "title": "Substitute back", "input": "-2\\left(\\frac{\\ln\\left|v+1\\right|}{2}-\\frac{\\ln\\left|v-1\\right|}{2}-v\\right)", "result": "=-2\\left(\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}-\\frac{\\sqrt{4-x^{2}}}{2}\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\frac{u}{2}$$", "result": "=-2\\left(\\frac{\\ln\\left|\\frac{u}{2}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{u}{2}-1\\right|}{2}-\\frac{u}{2}\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sqrt{4-x^{2}}$$", "result": "=-2\\left(\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}-\\frac{\\sqrt{4-x^{2}}}{2}\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "interim", "title": "Simplify $$-2\\left(\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}-\\frac{\\sqrt{4-x^{2}}}{2}\\right):{\\quad}-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{2}}$$", "input": "-2\\left(\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}-\\frac{\\sqrt{4-x^{2}}}{2}\\right)", "result": "=-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{2}}", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=\\left(-2\\right)\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}+\\left(-2\\right)\\left(-\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}\\right)+\\left(-2\\right)\\left(-\\frac{\\sqrt{4-x^{2}}}{2}\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a,\\:\\:\\left(-a\\right)\\left(-b\\right)=ab$$" ], "result": "=-2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}+2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}+2\\cdot\\:\\frac{\\sqrt{4-x^{2}}}{2}" }, { "type": "interim", "title": "Simplify $$-2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}+2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}+2\\cdot\\:\\frac{\\sqrt{4-x^{2}}}{2}:{\\quad}-\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|+\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|+\\sqrt{4-x^{2}}$$", "input": "-2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}+2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}+2\\cdot\\:\\frac{\\sqrt{4-x^{2}}}{2}", "result": "=-\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|+\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|+\\sqrt{4-x^{2}}", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}=\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|$$", "input": "2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}+1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvioLeycgw6B/tDi5B/+nN/HpL+FJi3g8BBB6PpY6x6MNgG9pClOQue52AYLVOWp3Q250WQvP2wkaKQ5hCmiePaOHMwViaLUXkeD+JukROhWdjZBdZkc0mXUcOIL4yXQQL2Uv4UmLeDwEEHo+ljrHow2Ab2kKU5C57nYBgtU5andDbP8vQyhiD4JSfqjIvcQ7tivZR7nZVCH2w7kTuRxAegbWusR3GGH+2/UWilK/eD4MM7LH0A7WZfDlyjdTEyiEO+l0FKddxJVcCVGb1P6EcOS9pRsG2zMlf0fHhxHoazclWL/85oFJFP2CDV+y8Lo6TM5BYY5RxCtG+LmTYDhwqoF1yK9yOuFDCUb5ZoQXQlwmL" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}=\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|$$", "input": "2\\cdot\\:\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\ln\\left|\\frac{\\sqrt{4-x^{2}}}{2}-1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvioLeycgw6B/tDi5B/+nN/HpL+FJi3g8BBB6PpY6x6MNggTdKL1+9NMHTR0toQlBsh50WQvP2wkaKQ5hCmiePaOHMwViaLUXkeD+JukROhWdjZBdZkc0mXUcOIL4yXQQL2Uv4UmLeDwEEHo+ljrHow2CBN0ovX700wdNHS2hCUGyHP8vQyhiD4JSfqjIvcQ7tivZR7nZVCH2w7kTuRxAegbWusR3GGH+2/UWilK/eD4MM7LH0A7WZfDlyjdTEyiEO+pT9gSFTgMyPw/IuF06i3GFpRsG2zMlf0fHhxHoazclWL/85oFJFP2CDV+y8Lo6TM6irgmINpjPmOARvM0vIFvpyK9yOuFDCUb5ZoQXQlwmL" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{\\sqrt{4-x^{2}}}{2}=\\sqrt{4-x^{2}}$$", "input": "2\\cdot\\:\\frac{\\sqrt{4-x^{2}}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{4-x^{2}}\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\sqrt{4-x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviuyx9AO1mXw5co3UxMohDvqdFkLz9sJGikOYQponj2jhzMFYmi1F5Hg/ibpEToVnY2zWP9Ngb5vmUP0A8KwBLz4/y9DKGIPglJ+qMi9xDu2K9lHudlUIfbDuRO5HEB6BtbrFWxRjxqjI9z4KIFBspYxUWbcrS3m3L3vJsg+2lE8vFkVDw5yEF2DEw4msQzKTew==" } }, { "type": "step", "result": "=-\\ln\\left|\\frac{\\sqrt{-x^{2}+4}}{2}+1\\right|+\\ln\\left|\\frac{\\sqrt{-x^{2}+4}}{2}-1\\right|+\\sqrt{-x^{2}+4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{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", "practiceLink": "/practice/integration-practice#area=main&subtopic=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}+1\\right|+\\ln\\left|\\frac{1}{2}\\sqrt{4-x^{2}}-1\\right|+\\sqrt{4-x^{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }