integral of 1/((x-1)sqrt(4x^2-8x+3))

{ "query": { "display": "$$\\int\\:\\frac{1}{\\left(x-1\\right)\\sqrt{4x^{2}-8x+3}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{(x-1)\\sqrt{4x^{2}-8x+3}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\arctan(\\sqrt{4x^{2}-8x+3})+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\left(x-1\\right)\\sqrt{4x^{2}-8x+3}}dx=\\arctan\\left(\\sqrt{4x^{2}-8x+3}\\right)+C$$", "input": "\\int\\:\\frac{1}{\\left(x-1\\right)\\sqrt{4x^{2}-8x+3}}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{\\left(x-1\\right)\\sqrt{4x^{2}-8x+3}}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: $$u=x-1$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x-1\\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(1\\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": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "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": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u\\sqrt{4x^{2}-8x+3}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u\\sqrt{4x^{2}-8x+3}}du" }, { "type": "interim", "title": "$$u=x-1\\quad\\Rightarrow\\quad\\:x=u+1$$", "input": "x-1=u", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=u", "result": "x=u+1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=u+1" }, { "type": "step", "primary": "Simplify", "result": "x=u+1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u\\sqrt{4\\left(u+1\\right)^{2}-8\\left(u+1\\right)+3}}du" }, { "type": "interim", "title": "Expand $$4\\left(u+1\\right)^{2}-8\\left(u+1\\right)+3:{\\quad}4u^{2}-1$$", "input": "4\\left(u+1\\right)^{2}-8\\left(u+1\\right)+3", "steps": [ { "type": "interim", "title": "$$\\left(u+1\\right)^{2}:{\\quad}u^{2}+2u+1$$", "result": "=4\\left(u^{2}+2u+1\\right)-8\\left(u+1\\right)+3", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=u,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=u^{2}+2u\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$u^{2}+2u\\cdot\\:1+1^{2}:{\\quad}u^{2}+2u+1$$", "input": "u^{2}+2u\\cdot\\:1+1^{2}", "result": "=u^{2}+2u+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=u^{2}+2\\cdot\\:1\\cdot\\:u+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=u^{2}+2u+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$4\\left(u^{2}+2u+1\\right):{\\quad}4u^{2}+8u+4$$", "input": "4\\left(u^{2}+2u+1\\right)", "result": "=4u^{2}+8u+4-8\\left(u+1\\right)+3", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=4u^{2}+4\\cdot\\:2u+4\\cdot\\:1", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "interim", "title": "Simplify $$4u^{2}+4\\cdot\\:2u+4\\cdot\\:1:{\\quad}4u^{2}+8u+4$$", "input": "4u^{2}+4\\cdot\\:2u+4\\cdot\\:1", "result": "=4u^{2}+8u+4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=4u^{2}+8u+4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4u^{2}+8u+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78yWjQug2Eeia1saMmORtmS061ljBSPJeENOw2efoSWu6+4pOCgOyFEg8oj3MGJE7/SUoe2xEEAhEKdn7CWeb803kCh3oevUunZ7/b0qFKBQwpI4T7Vi6VSfM5yZO+F5JIn9ODl79TNtC8BpzJOllTA==" } }, { "type": "interim", "title": "Expand $$-8\\left(u+1\\right):{\\quad}-8u-8$$", "input": "-8\\left(u+1\\right)", "result": "=4u^{2}+8u+4-8u-8+3", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-8,\\:b=u,\\:c=1$$" ], "result": "=-8u+\\left(-8\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-8u-8\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:1=8$$", "result": "=-8u-8" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NvtYD8Sp/WvMr/MhXR6mUAOfOVs9mPIqDLV5QIWwt3m63H3QtdSZy/2yp+LQ51lpgQUxJPyUNnGfVirkcwpVO75X8l7pvP3dqbB/LS8pZatFXUdUFeClGzCSixXIuTiA" } }, { "type": "interim", "title": "Simplify $$4u^{2}+8u+4-8u-8+3:{\\quad}4u^{2}-1$$", "input": "4u^{2}+8u+4-8u-8+3", "result": "=4u^{2}-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=4u^{2}+8u-8u+4-8+3" }, { "type": "step", "primary": "Add similar elements: $$8u-8u=0$$", "result": "=4u^{2}+4-8+3" }, { "type": "step", "primary": "Add/Subtract the numbers: $$4-8+3=-1$$", "result": "=4u^{2}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ea6haYnNZnfrzigmWi5UN+X9BqfksdI3bFxi65xCtQlwkKGJWEPFPk38sdJMsyPIR7Jw7V5tQwxLcQwsEVjlnPC30sSftAIFS6Qkpy19IkoPt3RvH3em6T49j66ClaPcX/QT0BtBCGGgLS6IqOe7bCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u\\sqrt{4u^{2}-1}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7wWwTTJ2Es7CpAkfxdGTBdMoZODrKa1TXdNigeYcSkwEym/NAi4LHcruw7BoRzkCzM3jmNT44FMtC1KZ4tDeWVW42mNeCByXqf6JNlkLL1HElghgaJDAHWD8s5U01/pFLQK5LSI6Q3uzWW0Ijs8RS5aKw4YpC51kvoY7QC6nje1dRSpN33oxZMojoqvYhvSJAFl6BLaDCHymr+DA1/BoNFsDA1Pfcj1+04Z2XKFwedeU" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u\\sqrt{4u^{2}-1}}du" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{u\\sqrt{4u^{2}-1}}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: $$v=\\sqrt{4u^{2}-1}$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=\\frac{4u}{\\sqrt{4u^{2}-1}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{4u^{2}-1}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{4u^{2}-1}}\\frac{d}{du}\\left(4u^{2}-1\\right)$$", "input": "\\frac{d}{du}\\left(\\sqrt{4u^{2}-1}\\right)", "result": "=\\frac{1}{2\\sqrt{4u^{2}-1}}\\frac{d}{du}\\left(4u^{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": [ "$$f=\\sqrt{v},\\:\\:v=4u^{2}-1$$" ], "result": "=\\frac{d}{dv}\\left(\\sqrt{v}\\right)\\frac{d}{du}\\left(4u^{2}-1\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(\\sqrt{v}\\right)=\\frac{1}{2\\sqrt{v}}$$", "input": "\\frac{d}{dv}\\left(\\sqrt{v}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{dv}\\left(v^{\\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}v^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}v^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{v}}$$", "input": "\\frac{1}{2}v^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{v}}", "steps": [ { "type": "interim", "title": "$$v^{\\frac{1}{2}-1}=v^{-\\frac{1}{2}}$$", "input": "v^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=v^{-\\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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+YHps///CwtnaIZ4m+tY4u0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjwqrM4HXYHBH4BvbM1QQD6wH2kDe5DGYTz3TrPquGdIjovvE/2xqO2EBVKbFFRMAlvlEplt2lwTz+GuNnTtBrmlQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}v^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$v^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{v}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{v}}", "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{v}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{v}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JiXymjtPdKrPsJFXCBCy2bH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJtaMF5l+Vtp+Br+G09+/YDIB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2ru6wQCVVKRqlBZr3/T5etYMTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{v}}\\frac{d}{du}\\left(4u^{2}-1\\right)" }, { "type": "step", "primary": "Substitute back $$v=4u^{2}-1$$", "result": "=\\frac{1}{2\\sqrt{4u^{2}-1}}\\frac{d}{du}\\left(4u^{2}-1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuE6PxbWnKi3vqKm3d2KLE/CQ0jRTB3XX2/mYqBTBQFV1FQ1qG6fJuWh1wKHeWVTZR/eiQ2bs0wqnHz5TvUcL8GBIQdF1r4bkEczkVETb80bg2zHzTE4agT2SP7LBdeMVMiz920OkzClDBMm895y8oaBL+zUVM/GvPXASgbzNAt/IaTy8gmKy5PY3el+RbUmlsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(4u^{2}-1\\right)=8u$$", "input": "\\frac{d}{du}\\left(4u^{2}-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{du}\\left(4u^{2}\\right)-\\frac{d}{du}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(4u^{2}\\right)=8u$$", "input": "\\frac{d}{du}\\left(4u^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{du}\\left(u^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmlNBeyYAxp6r2Y83eYMjyaTdaV09PMxEKZ9FieghTFwQFTal+18j5CaTaZWLMNoRqN6Hv6MoTMtvtU0IQwXdn/opg0pDqAhQx5eD06LEE3gvRWRQvqcpnBK6TL4xCiU8A==" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(1\\right)=0$$", "input": "\\frac{d}{du}\\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+idP5vLVrjUll6eMdYoonkTa3K+3a3OJF7IGZShVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTu3gZlAnOLgeYDts0TT4yEK" } }, { "type": "step", "result": "=8u-0" }, { "type": "step", "primary": "Simplify", "result": "=8u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{4u^{2}-1}}\\cdot\\:8u" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{4u^{2}-1}}\\cdot\\:8u:{\\quad}\\frac{4u}{\\sqrt{4u^{2}-1}}$$", "input": "\\frac{1}{2\\sqrt{4u^{2}-1}}\\cdot\\:8u", "result": "=\\frac{4u}{\\sqrt{4u^{2}-1}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:8u}{2\\sqrt{4u^{2}-1}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:8=8$$", "result": "=\\frac{8u}{2\\sqrt{4u^{2}-1}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{2}=4$$", "result": "=\\frac{4u}{\\sqrt{4u^{2}-1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13C79MVk3XNcHWgzsRTZsUQspWZXgxRFjoT7FJpeuj5s1NCUCWbkwGOY7PqKo3U/JLJdODbkdxph7FFl2Kda2lXRfWp2eziMfZK+IfjZxcUwjuP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXQ/xrrkgsn/brRjSw3V+PnXFnFlC5/quguysO8FKdJBKCRLgbN+3ckVN6LL9CUIMC8=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=\\frac{4u}{\\sqrt{4u^{2}-1}}du$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{\\sqrt{4u^{2}-1}}{4u}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{uv}\\cdot\\:\\frac{\\sqrt{4u^{2}-1}}{4u}dv" }, { "type": "step", "primary": "$$v=\\sqrt{4u^{2}-1}$$", "result": "=\\int\\:\\frac{1}{uv}\\cdot\\:\\frac{v}{4u}dv" }, { "type": "interim", "title": "Simplify $$\\frac{1}{uv}\\cdot\\:\\frac{v}{4u}:{\\quad}\\frac{1}{4u^{2}}$$", "input": "\\frac{1}{uv}\\cdot\\:\\frac{v}{4u}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:v}{uv\\cdot\\:4u}" }, { "type": "step", "primary": "Cancel the common factor: $$v$$", "result": "=\\frac{1}{u\\cdot\\:4u}" }, { "type": "interim", "title": "$$u\\cdot\\:4u=4u^{2}$$", "input": "u\\cdot\\:4u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=4u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=4u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79PBTGB1qx7QtdC6/kphdSyAn9lkDfZkicUGkO3EF+IrHdRORxbsyC2Nn3RC7++zeLfn1XZuzWrrulfH3irlcCsL8OVocQDnjfWwqTq4xygSwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{1}{4u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{4u^{2}}dv" }, { "type": "interim", "title": "$$v=\\sqrt{4u^{2}-1}\\quad\\Rightarrow\\quad\\:u^{2}=\\frac{v^{2}+1}{4}$$", "input": "\\sqrt{4u^{2}-1}=v", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}4u^{2}-1=v^{2}$$", "input": "\\sqrt{4u^{2}-1}=v", "result": "4u^{2}-1=v^{2}", "steps": [ { "type": "step", "result": "\\left(\\sqrt{4u^{2}-1}\\right)^{2}=v^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{4u^{2}-1}\\right)^{2}:{\\quad}4u^{2}-1$$", "input": "\\left(\\sqrt{4u^{2}-1}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(4u^{2}-1\\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(4u^{2}-1\\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": "=4u^{2}-1", "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+eNAieDpxyFrv2gq1a39O4EbySALAZlDhoAdFQF6AF4pPagvRKAKbro8EavzLN8GuIDU/PWVUjtAdd6EqSvsyX6r+5AezFilETfuNygjs0XPkV2hxyZSH7lGs/TcEChn4eo8RNxp01taCF72EuPKiync4fE=" } }, { "type": "step", "result": "4u^{2}-1=v^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QWJqV3Z2EuxLgtIcJ4rgDC0+uQTC2rJk2wnzJ36NyzAtVY86dFSmxhIBSkspfk/4gY2pwARdbpc94GT6N/QUat9Hql8XXPq6bNQlMm+36iNhmgLTdIEuzwdV+ksVN+7oK2hn7L5gTBUzF4g+U/sLR6Sg==" } }, { "type": "interim", "title": "Solve $$4u^{2}-1=v^{2}:{\\quad}u^{2}=\\frac{v^{2}+1}{4}$$", "input": "4u^{2}-1=v^{2}", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "4u^{2}-1=v^{2}", "result": "4u^{2}=v^{2}+1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "4u^{2}-1+1=v^{2}+1" }, { "type": "step", "primary": "Simplify", "result": "4u^{2}=v^{2}+1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4u^{2}=v^{2}+1", "result": "u^{2}=\\frac{v^{2}+1}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4u^{2}}{4}=\\frac{v^{2}}{4}+\\frac{1}{4}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{4u^{2}}{4}=\\frac{v^{2}}{4}+\\frac{1}{4}", "result": "u^{2}=\\frac{v^{2}+1}{4}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{4u^{2}}{4}:{\\quad}u^{2}$$", "input": "\\frac{4u^{2}}{4}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{4}=1$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I9nuUyPKVx+q9koICCpuDpWBb03HJU0DCUwYRjiC5hijkVi15I8rBefLi4Iyt2wrAbM3Da1Iig1s0xOw+GRJUYsaY2cs9YZ/08feWmaOgkirOT8NdezRd8zgGTcc8Q1vxpz3IBIvQoLTR/EOKGOiew==" } }, { "type": "interim", "title": "Simplify $$\\frac{v^{2}}{4}+\\frac{1}{4}:{\\quad}\\frac{v^{2}+1}{4}$$", "input": "\\frac{v^{2}}{4}+\\frac{1}{4}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{v^{2}+1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YxRMSKkW8Ycw0RIbgr9xhYW8COlu/fOM/LxWnlAaO2ItOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOFErdZy8DXUAKqCAc5ugIeOjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIOfODOfqL21ljQwU68B8ml5Toeo36lfEwKsZGa5BOmiNJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "u^{2}=\\frac{v^{2}+1}{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": "u^{2}=\\frac{v^{2}+1}{4}" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}u^{2}=\\frac{v^{2}+1}{4}\\:\\left\\{v\\ge\\:0\\right\\}$$", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\sqrt{4u^{2}-1}=v$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Plug$${\\quad}u^{2}=\\frac{v^{2}+1}{4}:{\\quad}\\sqrt{4\\frac{v^{2}+1}{4}-1}=v{\\quad}\\Rightarrow{\\quad}v\\ge\\:0$$", "input": "\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}=v", "steps": [ { "type": "interim", "title": "Square both sides:$${\\quad}v^{2}=v^{2}$$", "input": "\\sqrt{4\\cdot\\:\\frac{v^{2}+1}{4}-1}=v", "result": "v^{2}=v^{2}", "steps": [ { "type": "step", "result": "\\left(\\sqrt{4\\cdot\\:\\frac{v^{2}+1}{4}-1}\\right)^{2}=v^{2}" }, { "type": "interim", "title": "Expand $$\\left(\\sqrt{4\\frac{v^{2}+1}{4}-1}\\right)^{2}:{\\quad}v^{2}$$", "input": "\\left(\\sqrt{4\\cdot\\:\\frac{v^{2}+1}{4}-1}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(4\\frac{v^{2}+1}{4}-1\\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\\frac{v^{2}+1}{4}-1\\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\\frac{v^{2}+1}{4}-1", "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/" } }, { "type": "step", "primary": "Refine", "result": "=v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vz0OwVmz7FbkxdUH+eNAiWZj3AyibvkkQu6Of6Sb8lDPg6Rb7rAW3UsqV8MNX4Q1qv8t+8+OAov8Ad2CzUVHinUMUocDoN1vlETQDLubD2ulWMK6IC5dvUU+h2HdjSd9eqXxdc+rps1CUyb7fqI2GY5i0bX2psTZTw6KSw+CPpsxVCXESCfkWRHlX6uIUpYPpQBVKJ+sRpl53w48UztGJw==" } }, { "type": "step", "result": "v^{2}=v^{2}" } ], "meta": { "interimType": "Radicals Square Both Sides Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDaZCNCMlwFw9Zrpnfgo8QW86yXOpYYDpwhiyOt6Wy80TcOfrb/61zwK2LBdYWswiIOj6jZuF1csq1CYq9LN0VT+oZ4mKJkRwrWGK1DOyzmLKu34Rh5uB2zqsrT+/Z7gYbvbBmbuQNTF0TphKZ8RuvaEdpRXzE31YI2bc4eNnbBUDn9urK9SDnoRAVWzV4HVeI=" } }, { "type": "step", "primary": "Both sides are equal", "result": "\\mathrm{True\\:for\\:all}\\:v" }, { "type": "interim", "title": "Verify Solutions:$${\\quad}v<0\\:$$False$$,\\:\\:v=0\\:$$True$$,\\:\\:v>0\\:$$True", "input": "\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}=v", "steps": [ { "type": "step", "primary": "Combine domain interval with solution interval:", "result": "\\mathrm{True\\:for\\:all}\\:v" }, { "type": "interim", "title": "Find the function intervals:$${\\quad}v<0,\\:v=0,\\:v>0$$", "input": "\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}=v", "steps": [ { "type": "step", "primary": "Find the even roots arguments zeroes:" }, { "type": "interim", "title": "Solve $$4\\left(\\frac{v^{2}+1}{4}\\right)-1=0:{\\quad}v=0$$", "input": "4\\left(\\frac{v^{2}+1}{4}\\right)-1=0", "steps": [ { "type": "interim", "title": "Expand $$4\\left(\\frac{v^{2}+1}{4}\\right)-1:{\\quad}v^{2}$$", "input": "4\\left(\\frac{v^{2}+1}{4}\\right)-1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=4\\cdot\\:\\frac{v^{2}+1}{4}-1" }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{v^{2}+1}{4}=v^{2}+1$$", "input": "4\\cdot\\:\\frac{v^{2}+1}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(v^{2}+1\\right)\\cdot\\:4}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=v^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV8xBItd5234ZMqYgbf6Q2GN8gJ/ZZA32ZInFBpDtxBfiKlh1mi9YWktDdgmHzpMqwDj/L0MoYg+CUn6oyL3EO7YpFZM3rAAs8++TBGaiSNYNaAU13VmN9x5fk0jAsvPnbKmuyjkq9cAR+84MTlfGEpHg=" } }, { "type": "step", "result": "=v^{2}+1-1" }, { "type": "step", "primary": "$$1-1=0$$", "result": "=v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bmCOjrHGY8KwMwap9tYqhu1f4sad3g8nx//cxyBJ0pzeXCIFfkXoRCDVXhjJDXGVs7PdN8ocStDSJ08Eeyb0kcjP9vZe0h5cDZXk0KEZ9KgScn/M2sLYBGgxClqzCxM2PT6nhXeXs0GFNHkbl9X3Su0eMQIiGXhtj2GtGNlFPHg=" } }, { "type": "step", "result": "v^{2}=0" }, { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "v=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "v=0" }, { "type": "step", "primary": "The intervals are defined around the zeroes:", "result": "v<0,\\:v=0,\\:v>0" }, { "type": "step", "primary": "Combine intervals with domain", "result": "v<0,\\:v=0,\\:v>0" } ], "meta": { "interimType": "Abs Find Intervals Title 0Eq" } }, { "type": "step", "primary": "Check the solutions by plugging them into $$\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}=v$$<br/>Remove the ones that don't agree with the equation.", "secondary": [ "Plug$${\\quad}v<0:{\\quad}\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}\\ne\\:v{\\quad}\\Rightarrow{\\quad}$$False", "Plug$${\\quad}v=0:{\\quad}\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}=v{\\quad}\\Rightarrow{\\quad}$$True", "Plug$${\\quad}v>0:{\\quad}\\sqrt{4\\left(\\frac{v^{2}+1}{4}\\right)-1}=v{\\quad}\\Rightarrow{\\quad}$$True" ] } ], "meta": { "interimType": "Verifying Solutions Title 0Eq" } }, { "type": "step", "primary": "The solution is", "result": "v\\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": "u^{2}=\\frac{v^{2}+1}{4}\\:\\left\\{v\\ge\\:0\\right\\}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{4\\cdot\\:\\frac{v^{2}+1}{4}}dv" }, { "type": "interim", "title": "Multiply $$4\\cdot\\:\\frac{v^{2}+1}{4}\\::{\\quad}v^{2}+1$$", "input": "4\\cdot\\:\\frac{v^{2}+1}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(v^{2}+1\\right)\\cdot\\:4}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=v^{2}+1" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{v^{2}+1}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7wK5LSI6Q3uzWW0Ijs8RS5bAvNvQR+Fqrx1wwU4k2/sypN4cZPWgnwFqHQUcV4FHsfBLh5j/jJcd1Frv9s/1xSw0pWMfsJc1e/Z0+a/wFZqiVdawfdGtW4aUXHrK9DlWLTlqwrWSby5O1IGofAhPyurvbBmbuQNTF0TphKZ8RuvaR7hM6KADkFDngi1m57r+mR5n4zu7ICtqiQHGrVYWXPk=" } }, { "type": "step", "result": "=\\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": "=\\arctan\\left(v\\right)" }, { "type": "interim", "title": "Substitute back", "input": "\\arctan\\left(v\\right)", "result": "=\\arctan\\left(\\sqrt{4\\left(x-1\\right)^{2}-1}\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\sqrt{4u^{2}-1}$$", "result": "=\\arctan\\left(\\sqrt{4u^{2}-1}\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-1$$", "result": "=\\arctan\\left(\\sqrt{4\\left(x-1\\right)^{2}-1}\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "interim", "title": "Expand $$4\\left(x-1\\right)^{2}-1:{\\quad}4x^{2}-8x+3$$", "input": "4\\left(x-1\\right)^{2}-1", "steps": [ { "type": "interim", "title": "$$\\left(x-1\\right)^{2}:{\\quad}x^{2}-2x+1$$", "result": "=4\\left(x^{2}-2x+1\\right)-1", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\cdot\\:1+1^{2}:{\\quad}x^{2}-2x+1$$", "input": "x^{2}-2x\\cdot\\:1+1^{2}", "result": "=x^{2}-2x+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=x^{2}-2\\cdot\\:1\\cdot\\:x+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=x^{2}-2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$4\\left(x^{2}-2x+1\\right):{\\quad}4x^{2}-8x+4$$", "input": "4\\left(x^{2}-2x+1\\right)", "result": "=4x^{2}-8x+4-1", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=4x^{2}+4\\left(-2x\\right)+4\\cdot\\:1", "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": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=4x^{2}-4\\cdot\\:2x+4\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$4x^{2}-4\\cdot\\:2x+4\\cdot\\:1:{\\quad}4x^{2}-8x+4$$", "input": "4x^{2}-4\\cdot\\:2x+4\\cdot\\:1", "result": "=4x^{2}-8x+4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=4x^{2}-8x+4\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1=4$$", "result": "=4x^{2}-8x+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gF5HJyOcTpLd3D1PEO0Qri061ljBSPJeENOw2efoSWvmQlelvrtkN08zPFKZRRz9MAWApZj0PP4558ExBtsrCU3kCh3oevUunZ7/b0qFKBQSX8V1MWudy9VVKPP8vU2Stz0qitabHu7a7YyHuZwOsg==" } }, { "type": "step", "primary": "Add/Subtract the numbers: $$4-1=3$$", "result": "=4x^{2}-8x+3" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cb5Y2DEByw3p55v52Chaz1XTSum/z5kLpMzXS1UJIezCI3WNxGqYNpLhyslG+hzudMQ7RZeOzIrafdXehNaRH0DLjX7DcYtB54q3geRNejHUMPnSKxgsYPr0Brhz3ZLoa0mxWkA5wlaEQ6kNnRBpcA==" } }, { "type": "step", "result": "=\\arctan\\left(\\sqrt{4x^{2}-8x+3}\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\arctan\\left(\\sqrt{4x^{2}-8x+3}\\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=\\arctan(\\sqrt{4x^{2}-8x+3})+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }