integral of sin(-4x)

{ "query": { "display": "$$\\int\\:\\sin\\left(-4x\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\sin(-4x)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{4}\\cos(4x)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\sin\\left(-4x\\right)dx=\\frac{1}{4}\\cos\\left(4x\\right)+C$$", "input": "\\int\\:\\sin\\left(-4x\\right)dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\sin\\left(-4x\\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: $$u=-4x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-4$$", "input": "\\frac{d}{dx}\\left(-4x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-4\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=-4\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=-4", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpoS8cFIUsWpeKskwDy3gU6XIQHgliMhSOSNsNni19Ine1LhTd4QryVaorhCQ5aN9Q4bfwiV6iMLJ5sC1nL7dOYTWWQgRlNaEAF1qBQs3aBd" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-4dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-\\frac{1}{4}\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\sin\\left(u\\right)\\left(-\\frac{1}{4}\\right)du" }, { "type": "step", "result": "=\\int\\:-\\frac{1}{4}\\sin\\left(u\\right)du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7wApZgiOEg8e8CfgOYFPYC0rnvpYQ0aygJfymuicdqpZKVCPy+LyHQ1vNkAF21yRW7Ge8vt/bCNpz7wo+jCIC8qlPhnwBLQlal6krIDjtd2LWx4EPdRgTPU1CtyH0JjWC0UqTd96MWTKI6Kr2Ib0iQBZegS2gwh8pq/gwNfwaDRbAwNT33I9ftOGdlyhcHnXlA==" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{4}\\sin\\left(u\\right)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}{4}\\cdot\\:\\int\\:\\sin\\left(u\\right)du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\sin\\left(u\\right)du=-\\cos\\left(u\\right)$$", "result": "=-\\frac{1}{4}\\left(-\\cos\\left(u\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=-4x$$", "result": "=-\\frac{1}{4}\\left(-\\cos\\left(-4x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-\\frac{1}{4}\\left(-\\cos\\left(-4x\\right)\\right):{\\quad}\\frac{1}{4}\\cos\\left(4x\\right)$$", "input": "-\\frac{1}{4}\\left(-\\cos\\left(-4x\\right)\\right)", "result": "=\\frac{1}{4}\\cos\\left(4x\\right)", "steps": [ { "type": "step", "primary": "Use the negative angle identity: $$\\cos\\left(-x\\right)=\\cos\\left(x\\right)$$", "secondary": [], "result": "=-\\frac{1}{4}\\left(-\\cos\\left(4x\\right)\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1}{4}\\cos\\left(4x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e5SMbt6FBr+hdEROpxzom/U0jjFYYx+hdf6o99OBoPEgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ8yTlkamgHBge69GNXGvW1Sjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcINn1pYA9X6TjRfLcxv3zPWMLKXkL9wL6vNIgbpt/l5SA" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{4}\\cos\\left(4x\\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{1}{4}\\cos(4x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }