integral of sin^2(x)

{ "query": { "display": "integral $$\\sin^{2}\\left(x\\right)$$", "symbolab_question": "PRE_CALC#integral \\sin^{2}(x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{2}(x-\\frac{1}{2}\\sin(2x))+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\sin^{2}\\left(x\\right)dx=\\frac{1}{2}\\left(x-\\frac{1}{2}\\sin\\left(2x\\right)\\right)+C$$", "input": "\\int\\:\\sin^{2}\\left(x\\right)dx", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\sin^{2}\\left(x\\right)dx", "result": "=\\int\\:\\frac{1-\\cos\\left(2x\\right)}{2}dx", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\sin^{2}\\left(x\\right)=\\frac{1-\\cos\\left(2x\\right)}{2}$$", "result": "=\\int\\:\\frac{1-\\cos\\left(2x\\right)}{2}dx" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s776D0uVCQr2127QKBS9iLjP1fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7JFhxSzNcjgPzX10I7nnzbPxd9g5d3e6Rxwz/p45fiLkCWQYJBNrvQ2kgC5s4TyrECBBTEk/JQ2cZ9WKuRzClU7DcFgmo2GmeG2Kh9EaSPQP8xQp8yk2EdviTvfhz6ruO+9RQa+VFzMcO8uN2y3ZuWk" } }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:1-\\cos\\left(2x\\right)dx" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\frac{1}{2}\\left(\\int\\:1dx-\\int\\:\\cos\\left(2x\\right)dx\\right)" }, { "type": "interim", "title": "$$\\int\\:1dx=x$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\cos\\left(2x\\right)dx=\\frac{1}{2}\\sin\\left(2x\\right)$$", "input": "\\int\\:\\cos\\left(2x\\right)dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\cos\\left(2x\\right)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: $$u=2x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51iZ4v02Fm2dNqQJZnxCX4Je" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2}du$$" }, { "type": "step", "result": "=\\int\\:\\cos\\left(u\\right)\\frac{1}{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74q6JkKmzW6HevVc80KBrFssjvX7KVUO/AeCFSId4S33cZqb2ujmN2FEZC5M/msYIHiX35dQ/h01lIvxamZtt5Pfzl9vMjUAUvB5H3kSKKYo4Ruz4fIbA7DSu1jg0w5EAYEFMST8lDZxn1Yq5HMKVTsGLeMxVl55xMYxEfjKBatulcQUlMOhkqQvF9O8Q8/Z5g==" } }, { "type": "step", "result": "=\\int\\:\\cos\\left(u\\right)\\frac{1}{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": "=\\frac{1}{2}\\cdot\\:\\int\\:\\cos\\left(u\\right)du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\cos\\left(u\\right)du=\\sin\\left(u\\right)$$", "result": "=\\frac{1}{2}\\sin\\left(u\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=\\frac{1}{2}\\sin\\left(2x\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{2}\\left(x-\\frac{1}{2}\\sin\\left(2x\\right)\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{2}\\left(x-\\frac{1}{2}\\sin\\left(2x\\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", "practiceTopic": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{1}{2}(x-\\frac{1}{2}\\sin(2x))+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }