integral of (csc(x)cot(x))/(2-csc(x))

{ "query": { "display": "$$\\int\\:\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{2-\\csc\\left(x\\right)}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{\\csc(x)\\cot(x)}{2-\\csc(x)}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\ln\\left|2-\\csc(x)\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{2-\\csc\\left(x\\right)}dx=\\ln\\left|2-\\csc\\left(x\\right)\\right|+C$$", "input": "\\int\\:\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{2-\\csc\\left(x\\right)}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{2-\\csc\\left(x\\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: $$u=2-\\csc\\left(x\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\cot\\left(x\\right)\\csc\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(2-\\csc\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(2\\right)-\\frac{d}{dx}\\left(\\csc\\left(x\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(2\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\csc\\left(x\\right)\\right)=-\\cot\\left(x\\right)\\csc\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\csc\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\csc\\left(x\\right)\\right)=-\\cot\\left(x\\right)\\csc\\left(x\\right)$$", "result": "=-\\cot\\left(x\\right)\\csc\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnxvgjuCKmGvtiULPWqd/xX8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zIvS7vzr6uSegZMX95LfcDhkS3dlcCKpQTQcheuut7MkYxV7ATU2xUv3+xthyaPEpZvVLDxjkm2TPw3TECKxSR1X/2ShEyfoOfSdvdB2hVEj" } }, { "type": "step", "result": "=0-\\left(-\\cot\\left(x\\right)\\csc\\left(x\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\cot\\left(x\\right)\\csc\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\cot\\left(x\\right)\\csc\\left(x\\right)dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{\\cot\\left(x\\right)\\csc\\left(x\\right)}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{u}\\cdot\\:\\frac{1}{\\cot\\left(x\\right)\\csc\\left(x\\right)}du" }, { "type": "interim", "title": "Simplify $$\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{u}\\cdot\\:\\frac{1}{\\cot\\left(x\\right)\\csc\\left(x\\right)}:{\\quad}\\frac{1}{u}$$", "input": "\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)}{u}\\cdot\\:\\frac{1}{\\cot\\left(x\\right)\\csc\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\csc\\left(x\\right)\\cot\\left(x\\right)\\cdot\\:1}{u\\cot\\left(x\\right)\\csc\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\csc\\left(x\\right)$$", "result": "=\\frac{\\cot\\left(x\\right)\\cdot\\:1}{u\\cot\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\cot\\left(x\\right)$$", "result": "=\\frac{1}{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7//BjxgstAI27zyjZqzrzLYFQy/EH87+a/mGh+HA7JQyUIJIWPYRgLNOE4gKBJDQEnEjypn9AjSlgvJBiJApN19yrccDNZ+6MHaBthhc+pLHbHy9hmFUb8831ZbMI8TtmNtd1XhyuQh0R96pMGVsKFKjeh7+jKEzLb7VNCEMF3Z/OJPkjPbiuWYFSLF3ZysuiboLPDmLK3dLiupgaXBLgxY=" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=2-\\csc\\left(x\\right)$$", "result": "=\\ln\\left|2-\\csc\\left(x\\right)\\right|" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\ln\\left|2-\\csc\\left(x\\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#area=main&subtopic=Trig%20Power%20Multiplication", "practiceTopic": "Integral Trig Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\ln\\left|2-\\csc(x)\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }