integral of-3cos(2x)

{ "query": { "display": "$$\\int\\:-3\\cos\\left(2x\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int -3\\cos(2x)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{3}{2}\\sin(2x)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:-3\\cos\\left(2x\\right)dx=-\\frac{3}{2}\\sin\\left(2x\\right)+C$$", "input": "\\int\\:-3\\cos\\left(2x\\right)dx", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-3\\cdot\\:\\int\\:\\cos\\left(2x\\right)dx" }, { "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": "=-3\\cdot\\:\\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": "=-3\\cdot\\:\\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": "=-3\\cdot\\:\\frac{1}{2}\\sin\\left(u\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=-3\\cdot\\:\\frac{1}{2}\\sin\\left(2x\\right)" }, { "type": "interim", "title": "Simplify $$-3\\cdot\\:\\frac{1}{2}\\sin\\left(2x\\right):{\\quad}-\\frac{3}{2}\\sin\\left(2x\\right)$$", "input": "-3\\cdot\\:\\frac{1}{2}\\sin\\left(2x\\right)", "result": "=-\\frac{3}{2}\\sin\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:3}{2}\\sin\\left(2x\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-\\frac{3}{2}\\sin\\left(2x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yKyVpEFPWiPkRzHkKl2/ujqrHbwyajs8pox3tLU3DO4AlilG71elit3w1IBbYN0PlbtMv58phRb2qRbzdA+fvPEFJEVd7mITCLTlFGmw9X/6yHsCG4EHfPjZOAvk9Cx27kAjP76qW66lOUsURwT0nbpeehgYXnnxzR0N9doOX1Ou/JGQuRK/fqUfBKZsGpBREDsLwqmZIQ4ItAd5rnS5TQ==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{3}{2}\\sin\\left(2x\\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=-\\frac{3}{2}\\sin(2x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }