integral of 1/(sqrt(-x^2+4x+5))

{ "query": { "display": "$$\\int\\:\\frac{1}{\\sqrt{-x^{2}+4x+5}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{\\sqrt{-x^{2}+4x+5}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\arcsin(\\frac{1}{3}(x-2))+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\sqrt{-x^{2}+4x+5}}dx=\\arcsin\\left(\\frac{1}{3}\\left(x-2\\right)\\right)+C$$", "input": "\\int\\:\\frac{1}{\\sqrt{-x^{2}+4x+5}}dx", "steps": [ { "type": "interim", "title": "Complete the square $$-x^{2}+4x+5:{\\quad}-\\left(x-2\\right)^{2}+9$$", "input": "-x^{2}+4x+5", "steps": [ { "type": "step", "primary": "Write $$-x^{2}+4x+5\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-1$$" ], "result": "=-\\left(x^{2}-4x-5\\right)" }, { "type": "interim", "title": "$$2a=-4{\\quad:\\quad}a=-2$$", "input": "2a=-4", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-4", "result": "a=-2", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-4}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2a}{2}=\\frac{-4}{2}", "result": "a=-2", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2a}{2}:{\\quad}a$$", "input": "\\frac{2a}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=a" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n/9TIosDfOyZXH4usT9jsy061ljBSPJeENOw2efoSWuFFDuzzgIzA1F36c+SAFGgRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pQyGeXuoJHAaVrX92ZjOTT" } }, { "type": "interim", "title": "Simplify $$\\frac{-4}{2}:{\\quad}-2$$", "input": "\\frac{-4}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ej6kQOwlFwB+VchNYJxdtS061ljBSPJeENOw2efoSWvmnWoUV5J5qJ5JncpNUfjFo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCBY3WoCT90CjrhUQJNff6HUJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "a=-2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-2\\right)^{2}\\:$$", "result": "=-\\left(x^{2}-4x-5+\\left(-2\\right)^{2}-\\left(-2\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-4x+\\left(-2\\right)^{2}=\\left(x-2\\right)^{2}$$", "Complete the square" ], "result": "=-\\left(\\left(x-2\\right)^{2}-5-\\left(-2\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-\\left(x-2\\right)^{2}+9" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{-\\left(x-2\\right)^{2}+9}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{1}{\\sqrt{-\\left(x-2\\right)^{2}+9}}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: $$u=x-2$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "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", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{-u^{2}+9}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{-u^{2}+9}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73YBJdU3jgZIBfx9KjN2Y12wVS27wG3yCpM7qbjObAxHELUeKHRshXz/m/TuNH2EzOwprlSD9PwzJRbesBPZ/gKQiQ7cmY4R9W5sVTpBAI2Pb418gUEU1+xvbQxutNRs01sWBiqGaBJ/4OTHtDClr1H8lA/VN7Bzb69jg7eGqLYaeqXxdc+rps1CUyb7fqI2GSSO7MiB4lSDXTJsqCrSvba3EQh3uVlxId1MCwC+ZNkW" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{-u^{2}+9}}du" }, { "type": "interim", "title": "Apply Trigonometric Substitution", "input": "\\int\\:\\frac{1}{\\sqrt{-u^{2}+9}}du", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$" }, { "type": "step", "primary": "For $$\\sqrt{a-bx^2}\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}\\sin\\left(u\\right)$$<br/>$$a=9,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=3\\quad\\Rightarrow\\quad$$substitute $$x=3\\sin\\left(u\\right)$$" }, { "type": "interim", "title": "$$\\frac{du}{dv}=3\\cos\\left(v\\right)$$", "input": "\\frac{d}{dv}\\left(3\\sin\\left(v\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{d}{dv}\\left(\\sin\\left(v\\right)\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dv}\\left(\\sin\\left(v\\right)\\right)=\\cos\\left(v\\right)$$", "result": "=3\\cos\\left(v\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYvvo83mrHHVEDEdukbLZKoqQp7tdIFyr1eVqMMLZHDTGgeXEFCD58rDFDfybZf16JKc24kovR2uQyuCBmi2WICduTtmxMNEJep3u5NQFK4k7ok8zpLQhDjmeR8ctstIPzLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=3\\cos\\left(v\\right)dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}}\\cdot\\:3\\cos\\left(v\\right)dv" }, { "type": "interim", "title": "$$\\frac{1}{\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}}\\cdot\\:3\\cos\\left(v\\right)=1$$", "input": "\\frac{1}{\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}}\\cdot\\:3\\cos\\left(v\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}}=\\frac{1}{3\\sqrt{-\\sin^{2}\\left(v\\right)+1}}$$", "input": "\\frac{1}{\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}=3\\sqrt{-\\sin^{2}\\left(v\\right)+1}$$", "input": "\\sqrt{-\\left(3\\sin\\left(v\\right)\\right)^{2}+9}", "steps": [ { "type": "interim", "title": "$$\\left(3\\sin\\left(v\\right)\\right)^{2}=9\\sin^{2}\\left(v\\right)$$", "input": "\\left(3\\sin\\left(v\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=3^{2}\\sin^{2}\\left(v\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=9\\sin^{2}\\left(v\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I6MJmjNDQqHdgtfEtofWe3yRHuGw7+tM5METTDj6vVHgnyWPgw4gJm1U5dcKPWr1R+vJEk/6ZoTa1uHb4t/+nCKhOVCUCc2XtY98RHpl6aHGqWCS9mRE5XVD0W9MrQZsCRPLaD21VhKZQ79JqU1zQA==" } }, { "type": "step", "result": "=\\sqrt{-9\\sin^{2}\\left(v\\right)+9}" }, { "type": "interim", "title": "Factor $$-9\\sin^{2}\\left(v\\right)+9:{\\quad}9\\left(-\\sin^{2}\\left(v\\right)+1\\right)$$", "input": "-9\\sin^{2}\\left(v\\right)+9", "result": "=\\sqrt{9\\left(-\\sin^{2}\\left(v\\right)+1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-9\\sin^{2}\\left(v\\right)+9\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$9$$", "result": "=9\\left(-\\sin^{2}\\left(v\\right)+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\sqrt{9}\\sqrt{-\\sin^{2}\\left(v\\right)+1}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{9}=3$$", "input": "\\sqrt{9}", "result": "=3\\sqrt{-\\sin^{2}\\left(v\\right)+1}", "steps": [ { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74qA1izKjiq4Lmfry6aUMLbQNm8iq+o7BWz4Uy8xN9LogJ/ZZA32ZInFBpDtxBfiKIeivcAg3vKkJecw7zL/ItrdVszS7b4/ZAkoXn0pay8v/P/+v51eTuH2/F4MIu5mqIILQjqaZ85mk3SpRwGSV5Bx4CXk6XJigrLRwUmDHeozenX1x52qsImeJPn6T6+Ko6hEGhESjpCSIv4dDIM4r8Q==" } }, { "type": "step", "result": "=\\frac{1}{3\\sqrt{-\\sin^{2}\\left(v\\right)+1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGVzJEjQax2M+kaJdL0BCCAkPIW4ACtoPQlZdqQwGTTv7CUCWbkwGOY7PqKo3U/JLJd1qPFvk1EjyYKhta0IK7kBII6nSoZW1JPXCytN3b5m3I8TQhIFfWwQMmVwSLGafpP/829doud4baah5J7nhH2iTNTCAkV0IX+EM1D+/W5WeWh7IjxxEVM6ZEKySy7VcZ4zUld4O/cbryOZBenHgcPCNWwnQIIRI68OsEGF+GfZnsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{1}{3\\sqrt{-\\sin^{2}\\left(v\\right)+1}}\\cos\\left(v\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3\\cos\\left(v\\right)}{3\\sqrt{-\\sin^{2}\\left(v\\right)+1}}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{1\\cdot\\:\\cos\\left(v\\right)}{\\sqrt{-\\sin^{2}\\left(v\\right)+1}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(v\\right)=\\cos\\left(v\\right)$$", "result": "=\\frac{\\cos\\left(v\\right)}{\\sqrt{-\\sin^{2}\\left(v\\right)+1}}" }, { "type": "interim", "title": "Simplify $$\\sqrt{-\\sin^{2}\\left(v\\right)+1}:{\\quad}\\sqrt{\\cos^{2}\\left(v\\right)}$$", "input": "\\sqrt{-\\sin^{2}\\left(v\\right)+1}", "result": "=\\frac{\\cos\\left(v\\right)}{\\sqrt{\\cos^{2}\\left(v\\right)}}", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$1=\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)$$", "secondary": [ "$$1-\\sin^{2}\\left(x\\right)=\\cos^{2}\\left(x\\right)$$" ], "result": "=\\sqrt{\\cos^{2}\\left(v\\right)}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\sqrt{\\cos^{2}\\left(v\\right)}=\\cos\\left(v\\right)$$", "input": "\\sqrt{\\cos^{2}\\left(v\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=\\cos\\left(v\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wal2v4GOCNgSltOWkNdkuNg4GcxVtCzVPVf1Um1fEeerju+5Z51e/ZZSD3gRHwjBpceDSu0PZLy6G0bpJT0MLrtCR5dIjxQ5ASg+ZPFVSscu3zVWFHeNoYvtsRXFNKBdGE6lGSvs4ad4qkPABca6Xw==" } }, { "type": "step", "result": "=\\frac{\\cos\\left(v\\right)}{\\cos\\left(v\\right)}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGVzJEjQax2M+kaJdL0BCCAlPcs3aNA+ViOvfQrUULVcIZmQIkTIEAZaLpfDfR4kgSAlAlm5MBjmOz6iqN1PySyXNsbio4EtCJeiTO1yjhzmj5+1PCNrq4ENlfHWndC77EcWMkY1XY+cBrH4os53EHjcIfxFUT97Yd8GLIP49JHRwuzv1vVG/UXrQBzs57Mui+w==" } }, { "type": "step", "result": "=\\int\\:1dv" } ], "meta": { "interimType": "Integral Trig Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73YBJdU3jgZIBfx9KjN2Y11PqDl8w3KHK+6s8xMXplZySUM9pakkKILvT6Fs/PM359VEgBuKxZgOTX2ljN4yVJ9qQggBPzB4Qayiyi1+p6hLgSEHRda+G5BHM5FRE2/NG98f1deNovpIy/MNNl80RSuqkIX6kseCKdEth+cILnwypcFCaFWHZpLjalxP+vTj92ddx3j44O/DWa+NvwScI3iwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\int\\:1dv" }, { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:v" }, { "type": "interim", "title": "Substitute back", "input": "1\\cdot\\:v", "result": "=1\\cdot\\:\\arcsin\\left(\\frac{1}{3}\\left(x-2\\right)\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\arcsin\\left(\\frac{1}{3}u\\right)$$", "result": "=1\\cdot\\:\\arcsin\\left(\\frac{1}{3}u\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-2$$", "result": "=1\\cdot\\:\\arcsin\\left(\\frac{1}{3}\\left(x-2\\right)\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "step", "primary": "Simplify", "result": "=\\arcsin\\left(\\frac{1}{3}\\left(x-2\\right)\\right)", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\arcsin\\left(\\frac{1}{3}\\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=Trig%20Power%20Multiplication", "practiceTopic": "Integral Trig Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\arcsin(\\frac{1}{3}(x-2))+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }