integral of 1/(sqrt(12-x^2))

{ "query": { "display": "$$\\int\\:\\frac{1}{\\sqrt{12-x^{2}}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{\\sqrt{12-x^{2}}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\arcsin(\\frac{1}{2\\sqrt{3}}x)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\sqrt{12-x^{2}}}dx=\\arcsin\\left(\\frac{1}{2\\sqrt{3}}x\\right)+C$$", "input": "\\int\\:\\frac{1}{\\sqrt{12-x^{2}}}dx", "steps": [ { "type": "interim", "title": "Apply Trigonometric Substitution", "input": "\\int\\:\\frac{1}{\\sqrt{12-x^{2}}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$" }, { "type": "step", "primary": "For $$\\sqrt{a-bx^2}\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}\\sin\\left(u\\right)$$<br/>$$a=12,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=2\\sqrt{3}\\quad\\Rightarrow\\quad$$substitute $$x=2\\sqrt{3}\\sin\\left(u\\right)$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=2\\sqrt{3}\\cos\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\sqrt{3}\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$", "result": "=2\\sqrt{3}\\cos\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpP4IMJJCcikohWUKju2rpDdym0VJU+9mlcm2Cg6m8KI2RpLvcxZMcHtXUx/L8nG/ZXFaMxvkoyR9u+S439IJSRG9hLbQjwjoFQW6pLUmawHzVshnF+h2bdSxlx9kBmvi3IBi+PPSi7GQHW93wAqk4HnkuGZwmn2te9cXywyd4Uu" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=2\\sqrt{3}\\cos\\left(u\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}}\\cdot\\:2\\sqrt{3}\\cos\\left(u\\right)du" }, { "type": "interim", "title": "$$\\frac{1}{\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}}\\cdot\\:2\\sqrt{3}\\cos\\left(u\\right)=1$$", "input": "\\frac{1}{\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}}\\cdot\\:2\\sqrt{3}\\cos\\left(u\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}}=\\frac{1}{\\sqrt{12-12\\sin^{2}\\left(u\\right)}}$$", "input": "\\frac{1}{\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}=\\sqrt{12-12\\sin^{2}\\left(u\\right)}$$", "input": "\\sqrt{12-\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}=12\\sin^{2}\\left(u\\right)$$", "input": "\\left(2\\sqrt{3}\\sin\\left(u\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}\\left(\\sqrt{3}\\right)^{2}\\sin^{2}\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\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": "=3^{\\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": "=3", "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": "=2^{2}\\cdot\\:3\\sin^{2}\\left(u\\right)" }, { "type": "step", "primary": "Refine", "result": "=12\\sin^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Y6q7Xt710zlcWlCM2XluoXVefGEWxww82o4HMIrkBGDNGoPE9TME3q+OPmgkv2RQgw/Aa0GuOS807bvFVSMbC1qcq8rtojp+EWgcVAOZm8ROuLsZk5MlQVTG9bQ/x11e2NtKugsKjYHaZRrRbw/Y5UO2YFpSEB0sl3im82/YqB0=" } }, { "type": "step", "result": "=\\sqrt{12-12\\sin^{2}\\left(u\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SilInaeBOsQw1BND+4n4eRrhBY7shumTaWluk42y6tZdYiUSXWVPEp6viDrzMWEBcJChiVhDxT5N/LHSTLMjyJmmU6MMRJnHLKzwYLzuumdwbKahW4tX67Rmbd3Sbrphu0JHl0iPFDkBKD5k8VVKx112Ey1Q9sL6PuI940LIn04Z8Asywa+dxknD65UpQ3umF9mYa7/f3F9cs9VOuWvZJn9WzjIsqy4n27d0ZSil5E4=" } }, { "type": "step", "result": "=\\frac{1}{\\sqrt{12-12\\sin^{2}\\left(u\\right)}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGQA3sreFaAPdb6NQudRS/roukHtldAd+Jfbd1YyQWGHAAJYpRu9XpYrd8NSAW2DdD/KxLrO04AooUAReaJjhZCYOrFGrUjCOnoTbhyJzGuGEWA+dQCTmEkcuCH87M6poAf8//6/nV5O4fb8Xgwi7maq21p6kZ8iKyExH9dVbu0e0XXYTLVD2wvo+4j3jQsifTu7MWfO/ju1Z6llLxuX4ctt2ASXVN44GSAX8fSozdmNdH8FS9njlb2Pi+nFrUt5tg0zj0MZ3Is3d6GVhJxvp+8g=" } }, { "type": "step", "result": "=2\\sqrt{3}\\frac{1}{\\sqrt{-12\\sin^{2}\\left(u\\right)+12}}\\cos\\left(u\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2\\sqrt{3}\\cos\\left(u\\right)}{\\sqrt{12-12\\sin^{2}\\left(u\\right)}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2\\sqrt{3}\\cos\\left(u\\right)}{\\sqrt{12-12\\sin^{2}\\left(u\\right)}}" }, { "type": "interim", "title": "$$\\sqrt{12-12\\sin^{2}\\left(u\\right)}=2\\sqrt{3}\\cos\\left(u\\right)$$", "input": "\\sqrt{12-12\\sin^{2}\\left(u\\right)}", "steps": [ { "type": "interim", "title": "Simplify $$12-12\\sin^{2}\\left(u\\right):{\\quad}12\\cos^{2}\\left(u\\right)$$", "input": "12-12\\sin^{2}\\left(u\\right)", "result": "=\\sqrt{12\\cos^{2}\\left(u\\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": "=12\\cos^{2}\\left(u\\right)" }, { "type": "interim", "title": "Factor $$12-12\\sin^{2}\\left(u\\right):{\\quad}12\\left(1-\\sin^{2}\\left(u\\right)\\right)$$", "input": "12-12\\sin^{2}\\left(u\\right)", "result": "=12\\left(-\\sin^{2}\\left(u\\right)+1\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=12\\cdot\\:1-12\\sin^{2}\\left(u\\right)" }, { "type": "step", "primary": "Factor out common term $$12$$", "result": "=12\\left(1-\\sin^{2}\\left(u\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify 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{12}\\sqrt{\\cos^{2}\\left(u\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{12}=2\\sqrt{3}$$", "input": "\\sqrt{12}", "result": "=2\\sqrt{3}\\sqrt{\\cos^{2}\\left(u\\right)}", "steps": [ { "type": "interim", "title": "Prime factorization of $$12:{\\quad}2^{2}\\cdot\\:3$$", "input": "12", "result": "=\\sqrt{2^{2}\\cdot\\:3}", "steps": [ { "type": "step", "primary": "$$12\\:$$divides by $$2\\quad\\:12=6\\cdot\\:2$$", "result": "=2\\cdot\\:6" }, { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "result": "=2^{2}\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdzmDmiXg+V79OVxDmBC/OzVB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIECR4XWPWntAkaT/mGe0ZPFN" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{3}\\sqrt{2^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "secondary": [ "$$\\sqrt{\\cos^{2}\\left(u\\right)}=\\cos\\left(u\\right)$$" ], "result": "=2\\sqrt{3}\\cos\\left(u\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7r5/CLpxQ+QxhyvtoR94uFUUVBk8ReYKY+18fRcgp5tndd47a0hQ8flDbGsI5To1dlcVozG+SjJH275Ljf0glJEb2EttCPCOgVBbqktSZrAfMZAEaDOOFj3DLrZilrG7sH8FS9njlb2Pi+nFrUt5tg4vIohAx2sjPM6RojSGpUZnpoAyGNigWX6VPXGKrwl8+" } }, { "type": "step", "result": "=\\frac{2\\sqrt{3}\\cos\\left(u\\right)}{2\\sqrt{3}\\cos\\left(u\\right)}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGQA3sreFaAPdb6NQudRS/roukHtldAd+Jfbd1YyQWGHA+3TOpIuzRFGTrdsTmhkkOqvX3yCyeOQKvDfrGm3HylEDnzlbPZjyKgy1eUCFsLd5l1K1r0aaVT/E0zZakID9jP/829doud4baah5J7nhH2gc4KxDO7/xYJHQROSK0jS/86AmHosVP9QwPEjfDZoZFi8WYzFFCXrkvRMeRuuAEqNTSNb5I9Mk/QBhTeMmSVCqsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=\\int\\:1du" } ], "meta": { "interimType": "Integral Trig Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73YBJdU3jgZIBfx9KjN2Y12w7DX5Zc7+PsNwm853VIq+SUM9pakkKILvT6Fs/PM359VEgBuKxZgOTX2ljN4yVJ9qQggBPzB4Qayiyi1+p6hLgSEHRda+G5BHM5FRE2/NGzjSn9XikRqC1s1U3I1KpZCqkIX6kseCKdEth+cILnwypcFCaFWHZpLjalxP+vTj92ddx3j44O/DWa+NvwScI3iwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\int\\:1du" }, { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:u" }, { "type": "step", "primary": "Substitute back $$u=\\arcsin\\left(\\frac{1}{2\\sqrt{3}}x\\right)$$", "result": "=1\\cdot\\:\\arcsin\\left(\\frac{1}{2\\sqrt{3}}x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\arcsin\\left(\\frac{1}{2\\sqrt{3}}x\\right)", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\arcsin\\left(\\frac{1}{2\\sqrt{3}}x\\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}{2\\sqrt{3}}x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }