integral of 1/(xsqrt(x-2))

{ "query": { "display": "$$\\int\\:\\frac{1}{x\\sqrt{x-2}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{x\\sqrt{x-2}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\sqrt{2}\\arctan(\\sqrt{\\frac{1}{2}(x-2)})+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{x\\sqrt{x-2}}dx=\\sqrt{2}\\arctan\\left(\\sqrt{\\frac{1}{2}\\left(x-2\\right)}\\right)+C$$", "input": "\\int\\:\\frac{1}{x\\sqrt{x-2}}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{x\\sqrt{x-2}}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: $$u=\\sqrt{x-2}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\frac{1}{2\\sqrt{x-2}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x-2}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{x-2}}\\frac{d}{dx}\\left(x-2\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x-2}\\right)", "result": "=\\frac{1}{2\\sqrt{x-2}}\\frac{d}{dx}\\left(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": [ "$$f=\\sqrt{u},\\:\\:u=x-2$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(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(x-2\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-2$$", "result": "=\\frac{1}{2\\sqrt{x-2}}\\frac{d}{dx}\\left(x-2\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjasHTJ4Bo7yMP4qgzIlGEQ7VvR39GcOAKf0PPZxfBHNdLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi/484O9NJ/PDg7/ATYMA16brhJRF0rY2YSWbsai4br+SCVFJD8NzgqIuC8eLoJx97ohPPpYdnH1QfQQbEoEa1QQ3SmUvX4+ZxjvBdfdSGCsOwpcB1b+ayqTG1ksTw+VIa6/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-2\\right)=1$$", "input": "\\frac{d}{dx}\\left(x-2\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(2\\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+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{x-2}}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2\\sqrt{x-2}}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{1}{2\\sqrt{x-2}}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=2\\sqrt{x-2}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{xu}\\cdot\\:2\\sqrt{x-2}du" }, { "type": "step", "primary": "$$u=\\sqrt{x-2}$$", "result": "=\\int\\:\\frac{1}{xu}\\cdot\\:2udu" }, { "type": "interim", "title": "Simplify $$\\frac{1}{xu}\\cdot\\:2u:{\\quad}\\frac{2}{x}$$", "input": "\\frac{1}{xu}\\cdot\\:2u", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2u}{xu}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=\\frac{1\\cdot\\:2}{x}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{2}{x}du" }, { "type": "interim", "title": "$$u=\\sqrt{x-2}\\quad\\Rightarrow\\quad\\:x=u^{2}+2$$", "input": "\\sqrt{x-2}=u", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}x-2=u^{2}$$", "input": "\\sqrt{x-2}=u", "result": "x-2=u^{2}", "steps": [ { "type": "step", "result": "\\left(\\sqrt{x-2}\\right)^{2}=u^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{x-2}\\right)^{2}:{\\quad}x-2$$", "input": "\\left(\\sqrt{x-2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(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(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": "=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+eNAiXlZGWEQSUfHavgD2AJUl0SJ9HQfN3qOfAB4kBd9UOFDpA7clkQOa7zoZEIDggSG4WRLd2VwIqlBNByF6663syTD7/TDV+MdcR0GdN6+1FDtX7CkT/M/Z3Npt1fRcedo+ompXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "x-2=u^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWts2XHEiSFUHmb7lyFu4Mlqara+52Td5xcUnwkQYXwhQAiVyPqAZh7yDlcQYf8OCnA8cNB8Ab3wAU7PsKnISMSpdru72i1HipIeOo3/2JsX1TmSqfo3P/KqBfD7c9O7QlJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Solve $$x-2=u^{2}:{\\quad}x=u^{2}+2$$", "input": "x-2=u^{2}", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2=u^{2}", "result": "x=u^{2}+2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2=u^{2}+2" }, { "type": "step", "primary": "Simplify", "result": "x=u^{2}+2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=u^{2}+2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{2}{u^{2}+2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/5jm/KimRuzD32ZGz9HYLVKrQH0z4nmb2cbVBF1rNb11FQ1qG6fJuWh1wKHeWVTZXKUPM+MYGzicg9RPlX4KpSBIQdF1r4bkEczkVETb80bSjg2M5bm4F1szvaryX0YooVAoEKTZWGL3JfUWIXU2lZN5Aod6Hr1Lp2e/29KhSgULaSrnXlspVqKDNgDo1fDttD7qPU27hAgAheGDuF9tCk=" } }, { "type": "step", "result": "=\\int\\:\\frac{2}{u^{2}+2}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": "=2\\cdot\\:\\int\\:\\frac{1}{u^{2}+2}du" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{u^{2}+2}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: $$u=\\sqrt{2}v$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=2,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{2}\\quad\\Rightarrow\\quad$$substitute $$x=\\sqrt{2}u$$" }, { "type": "interim", "title": "$$\\frac{du}{dv}=\\sqrt{2}$$", "input": "\\frac{d}{dv}\\left(\\sqrt{2}v\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\sqrt{2}\\frac{dv}{dv}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=\\sqrt{2}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\sqrt{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgaidu57Pbe8dZdeFPTkKL8uQWjpWyJA7BQvvk8gDZbeo5FYteSPKwXny4uCMrdsK/2bzTYkF7lzP90P9Vne/lJTW26qciuyUBGXQExCUedY0G/lubLXvwp/TGk0YovnTYmpXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\sqrt{2}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\left(\\sqrt{2}v\\right)^{2}+2}\\sqrt{2}dv" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\left(\\sqrt{2}v\\right)^{2}+2}\\sqrt{2}:{\\quad}\\frac{1}{\\sqrt{2}\\left(v^{2}+1\\right)}$$", "input": "\\frac{1}{\\left(\\sqrt{2}v\\right)^{2}+2}\\sqrt{2}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\left(\\sqrt{2}v\\right)^{2}+2}=\\frac{1}{2v^{2}+2}$$", "input": "\\frac{1}{\\left(\\sqrt{2}v\\right)^{2}+2}", "steps": [ { "type": "interim", "title": "$$\\left(\\sqrt{2}v\\right)^{2}=2v^{2}$$", "input": "\\left(\\sqrt{2}v\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=\\left(\\sqrt{2}\\right)^{2}v^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{2}\\right)^{2}:{\\quad}2$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\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": "=2^{\\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": "=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": { "interimType": "N/A" } }, { "type": "step", "result": "=2v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xOIoEFKOyXO4sKNu867CJlnyYRz18HvB+rp63mPitc+zsHBJV0oRhKqf7h8oBCga5pvhQTt+B84TVyWim9EWZs8StGii13rXKD9EwInsiycbV4cg51jiKuEbLe0Q6Pf6" } }, { "type": "step", "result": "=\\frac{1}{2v^{2}+2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fhdIr33/zTx3qSUMQVzS8C2SZTeTox7dQHpapp+uFF4tOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOA177gEeTRAya+QQVqkhH/1kS3dlcCKpQTQcheuut7MkzhUp/haUDSLLXTxIcFOKmO2llyyHNyeUI8Rfsrzu6exeGj0Ax/17nNngTVMxvBe2sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=\\sqrt{2}\\frac{1}{2v^{2}+2}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{2}}{2v^{2}+2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sqrt{2}=\\sqrt{2}$$", "result": "=\\frac{\\sqrt{2}}{2v^{2}+2}" }, { "type": "interim", "title": "Factor $$2v^{2}+2:{\\quad}2\\left(v^{2}+1\\right)$$", "input": "2v^{2}+2", "result": "=\\frac{\\sqrt{2}}{2\\left(v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2v^{2}+2\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(v^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{2}}{2\\left(v^{2}+1\\right)}:{\\quad}\\frac{1}{\\sqrt{2}\\left(v^{2}+1\\right)}$$", "input": "\\frac{\\sqrt{2}}{2\\left(v^{2}+1\\right)}", "result": "=\\frac{1}{\\sqrt{2}\\left(v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{\\frac{1}{2}}}{2\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{2^{\\frac{1}{2}}}{2^{1}}=\\frac{1}{2^{1-\\frac{1}{2}}}$$" ], "result": "=\\frac{1}{2^{-\\frac{1}{2}+1}\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{1}{2^{\\frac{1}{2}}\\left(v^{2}+1\\right)}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$2^{\\frac{1}{2}}=\\sqrt{2}$$" ], "result": "=\\frac{1}{\\sqrt{2}\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjxskgRRYHBi1Thc3A3et5jREvW/3q6F0tfkBFmaqQ8SCUCWbkwGOY7PqKo3U/JLJbDue9wVrd71EaduDvi1olLnhM2cJyA9KXXrJLyQcGdaZEt3ZXAiqUE0HIXrrrezJEAWxP1tUPfnVZd59cq9pwAussnnnisJndCAm4z+zno8TSLFcygAzOaJ9v+FHfp7YQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{2}\\left(v^{2}+1\\right)}dv" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=2\\cdot\\:\\int\\:\\frac{1}{\\sqrt{2}\\left(v^{2}+1\\right)}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\\:\\frac{1}{\\sqrt{2}}\\cdot\\:\\int\\:\\frac{1}{v^{2}+1}dv" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{v^{2}+1}dv=\\arctan\\left(v\\right)$$", "result": "=2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(v\\right)" }, { "type": "interim", "title": "Substitute back", "input": "2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(v\\right)", "result": "=2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(\\frac{\\sqrt{x-2}}{\\sqrt{2}}\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\frac{u}{\\sqrt{2}}$$", "result": "=2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(\\frac{u}{\\sqrt{2}}\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sqrt{x-2}$$", "result": "=2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(\\frac{\\sqrt{x-2}}{\\sqrt{2}}\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(\\frac{\\sqrt{x-2}}{\\sqrt{2}}\\right):{\\quad}\\sqrt{2}\\arctan\\left(\\sqrt{\\frac{1}{2}\\left(x-2\\right)}\\right)$$", "input": "2\\cdot\\:\\frac{1}{\\sqrt{2}}\\arctan\\left(\\frac{\\sqrt{x-2}}{\\sqrt{2}}\\right)", "result": "=\\sqrt{2}\\arctan\\left(\\sqrt{\\frac{1}{2}\\left(x-2\\right)}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{\\sqrt{2}}\\arctan\\left(\\frac{\\sqrt{x-2}}{\\sqrt{2}}\\right)" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:2}{\\sqrt{2}}=\\sqrt{2}$$", "input": "\\frac{1\\cdot\\:2}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{\\sqrt{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2}{2^{\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{2^{1}}{2^{\\frac{1}{2}}}=2^{1-\\frac{1}{2}}$$" ], "result": "=2^{1-\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=2^{\\frac{1}{2}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$2^{\\frac{1}{2}}=\\sqrt{2}$$" ], "result": "=\\sqrt{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCnlQAnPxIHPVTJhVgj7kPzdV00rpv8+ZC6TM10tVCSHsoDcnz6I6EbFIZGDoyqUAxyG0fRs8j/ot8KSXfbmontwGe2lWLfSPMfEeyVlAtmsQM5N0gkEVETMEj6TFp18akRY/IcyQ9jGcf2XIWN4f6q8=" } }, { "type": "step", "result": "=\\sqrt{2}\\arctan\\left(\\frac{\\sqrt{x-2}}{\\sqrt{2}}\\right)" }, { "type": "interim", "title": "$$\\frac{\\sqrt{x-2}}{\\sqrt{2}}=\\sqrt{\\frac{x-2}{2}}$$", "input": "\\frac{\\sqrt{x-2}}{\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Combine same powers : $$\\frac{\\sqrt{x}}{\\sqrt{y}}=\\sqrt{\\frac{x}{y}}$$", "result": "=\\sqrt{\\frac{x-2}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMeuHUbnyd3/MzKGlVg4lxkcAlilG71elit3w1IBbYN0PTHz/8WsQwxEczVud8iApq/W0kJjqL6FrQ9pjJTUTw/pqzjWG0dFM/TWnjzf0PnlQ/LCYFUGiVDrbRo1tMF1Z65HRtHj3o1RE9vc1v+Z6z6AO+7cUlNIlDJ2Ds29Lf8Nip3p4tHJlpEA9I2lAiL/TqA==" } }, { "type": "step", "result": "=\\sqrt{2}\\arctan\\left(\\sqrt{\\frac{x-2}{2}}\\right)" }, { "type": "step", "result": "=\\arctan\\left(\\sqrt{\\frac{1}{2}\\left(x-2\\right)}\\right)\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvis7mY3sDWWBdU4t1pQzhuRcM1nHS4WFVxWmZMNMXSlhTaajlv+xkY15LscaizZU0vEBtAOukGnVgBfS3sE5bbO6rju+5Z51e/ZZSD3gRHwjByh2RNvOunPza7Tyry/YY+L5KNHeb80iuosIYCEkodjRwF7+7sWJhAUVDujrZ39KkZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2UQNuZdvAUkLju+GWp/hefzuZjewNZYF1Ti3WlDOG5FwzWcdLhYVXFaZkw0xdKWFNpqOW/7GRjXkuxxqLNlTS8X6i/EHQxUZEnJrsvAqTkmg==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\sqrt{2}\\arctan\\left(\\sqrt{\\frac{1}{2}\\left(x-2\\right)}\\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=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\sqrt{2}\\arctan(\\sqrt{\\frac{1}{2}(x-2)})+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }