integral of (arcsin(x))/((1-x^2)^{1/2)}

{ "query": { "display": "$$\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{\\arcsin(x)}{(1-x^{2})^{\\frac{1}{2}}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\arcsin^{2}(x)-\\frac{\\arcsin^{2}(x)}{2}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}dx=\\arcsin^{2}\\left(x\\right)-\\frac{\\arcsin^{2}\\left(x\\right)}{2}+C$$", "input": "\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}dx", "steps": [ { "type": "interim", "title": "Apply Integration By Parts", "input": "\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}dx", "steps": [ { "type": "definition", "title": "Integration By Parts definition", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=\\arcsin\\left(x\\right)$$" }, { "type": "step", "primary": "$$v'=\\frac{1}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}$$" }, { "type": "interim", "title": "$$u'=\\frac{d}{dx}\\left(\\arcsin\\left(x\\right)\\right)=\\frac{1}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(\\arcsin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\arcsin\\left(x\\right)\\right)=\\frac{1}{\\sqrt{1-x^{2}}}$$", "result": "=\\frac{1}{\\sqrt{1-x^{2}}}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoNIWrQwDL8Ty+Lo4qSfhfLgLLGDwG8USUPGEfybPT5wqKwXnDyHJSOk7SW/uMHpmJCkbpQEWj02u4uNNwgketI00bQ3nrKyJHL8f9aiPEoqJmIz4BZtOGqrlPIp2uiVBizNTMzNFldKnO6duftKuFIrVdbLOOnWeyjBdCUXMslAE0OEBYP9ypDNUmNpOVQn9RI75SVGEmIF4gGtjSpRl20=" } }, { "type": "interim", "title": "$$v=\\int\\:\\frac{1}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}dx=\\arcsin\\left(x\\right)$$", "input": "\\int\\:\\frac{1}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}dx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{\\left(1-x^{2}\\right)^{\\frac{1}{2}}}=\\left(1-x^{2}\\right)^{-\\frac{1}{2}}$$" ], "result": "=\\int\\:\\left(1-x^{2}\\right)^{-\\frac{1}{2}}dx", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Apply Trigonometric Substitution", "input": "\\int\\:\\left(1-x^{2}\\right)^{-\\frac{1}{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/>$$a=1,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=1\\quad\\Rightarrow\\quad$$substitute $$x=\\sin\\left(u\\right)$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=\\cos\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$", "result": "=\\cos\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgerJLn9ae0g0/tUjnRuL1v8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugHqnJiEuQ8NpaCSOBx7rI7c9lxxWegcX2h0jAOpU/pWBOXdLN4kU1Rsu99QYXFxww==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\cos\\left(u\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\left(1-\\sin^{2}\\left(u\\right)\\right)^{-\\frac{1}{2}}\\cos\\left(u\\right)du" }, { "type": "interim", "title": "$$\\left(1-\\sin^{2}\\left(u\\right)\\right)^{-\\frac{1}{2}}\\cos\\left(u\\right)=1$$", "input": "\\left(1-\\sin^{2}\\left(u\\right)\\right)^{-\\frac{1}{2}}\\cos\\left(u\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$\\left(1-\\sin^{2}\\left(u\\right)\\right)^{-\\frac{1}{2}}=\\frac{1}{\\left(1-\\sin^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}}$$" ], "result": "=\\frac{1}{\\left(-\\sin^{2}\\left(u\\right)+1\\right)^{\\frac{1}{2}}}\\cos\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\cos\\left(u\\right)}{\\left(1-\\sin^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(u\\right)=\\cos\\left(u\\right)$$", "result": "=\\frac{\\cos\\left(u\\right)}{\\left(1-\\sin^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}}" }, { "type": "interim", "title": "Simplify $$\\left(1-\\sin^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}:{\\quad}\\left(\\cos^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}$$", "input": "\\left(1-\\sin^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}", "result": "=\\frac{\\cos\\left(u\\right)}{\\left(\\cos^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}}", "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": "=\\left(\\cos^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\left(\\cos^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}=\\cos\\left(u\\right)$$", "input": "\\left(\\cos^{2}\\left(u\\right)\\right)^{\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad$$ assuming $$a\\ge0$$", "result": "=\\cos^{2\\cdot\\:\\frac{1}{2}}\\left(u\\right)" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{2}=1$$", "input": "2\\cdot\\:\\frac{1}{2}", "result": "=\\cos\\left(u\\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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviunYaDZZLRLMVMbjt+IYzOarju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nTQhqRfFhpwnpyuz2TS2M3xMz+u325qtellRtF+nqNeo" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nqXADW5KIJmIiHZ3YOdDL/FTUE1bel/SLLvL3mMgA/MtOtZYwUjyXhDTsNnn6ElrZq9tm84rEMEKfeYk5PQhZreZeL/E8IVmb734Dfu7ii6hn9nlZuea2Jx3jtYU2NjKcYtr4y3wTL1NkyWksjnETwvkJ+BFZeo/ouWernBbVwM=" } }, { "type": "step", "result": "=\\frac{\\cos\\left(u\\right)}{\\cos\\left(u\\right)}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jfNBjkAuRlMA2s7HhfxusY0Zqs54tJNWlTyADUrZQYy89sbuuoTLGbrHjCkMUHUCzRqDxPUzBN6vjj5oJL9kUB3Hulx3XH9sBakgvyECKe3c2Yf0+UubXi1t2zntW9J80IioVrnpaM/nbZ9mKp7CLBJFXYxzPsstPE/MC7FDP3kisjIOOHvaeRuxkXydnp/Y" } }, { "type": "step", "result": "=\\int\\:1du" } ], "meta": { "interimType": "Integral Trig Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zkdD/Puuox65ayOgI0vCmM88fRKv5w90U6neulws3hELI71+ylVDvwHghUiHeEt93cdDbGQhM4u2I3IuvMhSrGkCgntDFdUQWCp8pL+1z0BPbm+EfPW2lf+ZDdfktozvYkqgwZf+B/680vtknxImH1K28PwYoM0kk3yqK7RdFxV1jHSxmhzEZ9PLkIQK5KP7s40saExuKBxnPZJToEFRA4kt3WiGR7ZaCaXvz77bMjS" } }, { "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 $$u=\\arcsin\\left(x\\right)$$", "result": "=1\\cdot\\:\\arcsin\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\arcsin\\left(x\\right)", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\arcsin\\left(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", "interimType": "Integrals" } }, { "type": "step", "result": "=\\arcsin\\left(x\\right)\\arcsin\\left(x\\right)-\\int\\:\\frac{1}{\\sqrt{1-x^{2}}}\\arcsin\\left(x\\right)dx" }, { "type": "interim", "title": "Simplify", "input": "\\arcsin\\left(x\\right)\\arcsin\\left(x\\right)-\\int\\:\\frac{1}{\\sqrt{1-x^{2}}}\\arcsin\\left(x\\right)dx", "result": "=\\arcsin^{2}\\left(x\\right)-\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}dx", "steps": [ { "type": "interim", "title": "$$\\arcsin\\left(x\\right)\\arcsin\\left(x\\right)=\\arcsin^{2}\\left(x\\right)$$", "input": "\\arcsin\\left(x\\right)\\arcsin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\arcsin\\left(x\\right)\\arcsin\\left(x\\right)=\\:\\arcsin^{1+1}\\left(x\\right)$$" ], "result": "=\\arcsin^{1+1}\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\arcsin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IMLFIyEGSvYXmgoJNl+OmrETM/+uhTmDHHA0P8rZApd1g99dC9fj9sg0EHzBIRDRenX1+SbnGgVZYG/NegyLPrSNzHBzn+u0L3aBaPmXlnL3aVcQjiUZ4KahkpmglMImxIlIBNcZbIfkZXoxpIhP57E6evUjOOX+6UeZDi8st+Y=" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\sqrt{1-x^{2}}}\\arcsin\\left(x\\right)dx=\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}dx$$", "input": "\\int\\:\\frac{1}{\\sqrt{1-x^{2}}}\\arcsin\\left(x\\right)dx", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{\\sqrt{1-x^{2}}}\\arcsin\\left(x\\right)\\::{\\quad}\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{1}{\\sqrt{1-x^{2}}}\\arcsin\\left(x\\right)", "result": "=\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}dx", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\arcsin\\left(x\\right)=\\arcsin\\left(x\\right)$$", "result": "=\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+zjj+OTpTxUQ+V5n/tUrpYMTgbeeC0juQZMjBQOTgKLNVeUYKM/hA1cY3MTwZJjnLTrWWMFI8l4Q07DZ5+hJa0gaP/jugaZSbcVO47OXf3CyHQcxRz6MtnYgWUtZ9M+PeSx9TJT05rhvTrpG0+umafcodjPBcYMtWZm+ngqQDbPcqisQD9DQJsxv98F/mvnky+I7oWVIjnPsglv/bFpPOiKe4n6DbNuGkXSABiGpCa/mI//OWDXFYRns+p4Rj72GyFBU+k8FJuTt12zLKUAUmr+h80jI05W2Ml9hYG2s7bu6DjtD2YPgMamSYo3yJlgM" } }, { "type": "step", "result": "=\\arcsin^{2}\\left(x\\right)-\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{-x^{2}+1}}dx" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zYPzinASao5ZOlFdiAfX/mDGnRDpU4JsEEs+V/IjaYe4IPah6UIyXl1jRZO84flsJN1pXT08zEQpn0WJ6CFMXCBNb/m5yrxwgYreZuaw8Pcucp6Rb72YkISmpQBDTvfIuKVnrMmLkPc7l0+8CrMMqOmhEE14t7egapxzCiJgeIcNg/OKcBJqjlk6UV2IB9f+R0VSdPHoSLgTYmuh125dYwNIbeU3LSJiVjwn7wUN31K5sNavoYxRIysyUYGC00Rkj62IHHS87J6eeFxI5gXHUvvYR7LkUjebw5zGpznhuan" } }, { "type": "step", "result": "=\\arcsin^{2}\\left(x\\right)-\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}dx=\\frac{\\arcsin^{2}\\left(x\\right)}{2}$$", "input": "\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-x^{2}}}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\arcsin\\left(x\\right)}{\\sqrt{1-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: $$u=\\arcsin\\left(x\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\frac{1}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(\\arcsin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\arcsin\\left(x\\right)\\right)=\\frac{1}{\\sqrt{1-x^{2}}}$$", "result": "=\\frac{1}{\\sqrt{1-x^{2}}}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoNIWrQwDL8Ty+Lo4qSfhfLgLLGDwG8USUPGEfybPT5wqKwXnDyHJSOk7SW/uMHpmJCkbpQEWj02u4uNNwgketI00bQ3nrKyJHL8f9aiPEoqDht/CJXqIwsnmwLWcvt05tlDXwj5y8M8OnUxE093OWy/X+I3tUIB3OSMChoYnFHpmBnwXJJL5Awm1ZDb0OOTxw==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{1}{\\sqrt{1-x^{2}}}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\sqrt{1-x^{2}}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{u}{\\sqrt{1-x^{2}}}\\sqrt{1-x^{2}}du" }, { "type": "interim", "title": "Simplify $$\\frac{u}{\\sqrt{1-x^{2}}}\\sqrt{1-x^{2}}:{\\quad}u$$", "input": "\\frac{u}{\\sqrt{1-x^{2}}}\\sqrt{1-x^{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{u\\sqrt{1-x^{2}}}{\\sqrt{1-x^{2}}}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{1-x^{2}}$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:udu" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zYPzinASao5ZOlFdiAfX/kdFUnTx6Ei4E2JrodduXWM10zYa+Xk9+cnpsO7HwxFrnEjypn9AjSlgvJBiJApN19yrccDNZ+6MHaBthhc+pLHbHy9hmFUb8831ZbMI8TtmKZZK35D7tNfkSrVgBVU6fYEuDOVaQvKofqHoY5jNapscN6G3wC0aNOpX3GMtOsWhImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=\\int\\:udu" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:udu", "result": "=\\frac{u^{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{1+1}}{1+1}:{\\quad}\\frac{u^{2}}{2}$$", "input": "\\frac{u^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77irOeniMfrJKKN+TrhAcvL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpAUgTzPSdH5PWV4NCtvwjA7/YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } }, { "type": "step", "primary": "Substitute back $$u=\\arcsin\\left(x\\right)$$", "result": "=\\frac{\\arcsin^{2}\\left(x\\right)}{2}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\arcsin^{2}\\left(x\\right)-\\frac{\\arcsin^{2}\\left(x\\right)}{2}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\arcsin^{2}\\left(x\\right)-\\frac{\\arcsin^{2}\\left(x\\right)}{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=Trig%20Power%20Multiplication", "practiceTopic": "Integral Trig Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\arcsin^{2}(x)-\\frac{\\arcsin^{2}(x)}{2}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }