integral of x+2/(sin^2(x))

{ "query": { "display": "$$\\int\\:x+\\frac{2}{\\sin^{2}\\left(x\\right)}dx$$", "symbolab_question": "BIG_OPERATOR#\\int x+\\frac{2}{\\sin^{2}(x)}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{x^{2}}{2}-2\\cot(x)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:x+\\frac{2}{\\sin^{2}\\left(x\\right)}dx=\\frac{x^{2}}{2}-2\\cot\\left(x\\right)+C$$", "input": "\\int\\:x+\\frac{2}{\\sin^{2}\\left(x\\right)}dx", "steps": [ { "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": "=\\int\\:xdx+\\int\\:\\frac{2}{\\sin^{2}\\left(x\\right)}dx" }, { "type": "interim", "title": "$$\\int\\:xdx=\\frac{x^{2}}{2}$$", "input": "\\int\\:xdx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=\\frac{x^{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{x^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{1+1}}{1+1}:{\\quad}\\frac{x^{2}}{2}$$", "input": "\\frac{x^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7814/6/Jz6acDoAMznrJ9GL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpPzIcDl+e6/8g9uDsiVdOq//YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{2}{\\sin^{2}\\left(x\\right)}dx=-2\\cot\\left(x\\right)$$", "input": "\\int\\:\\frac{2}{\\sin^{2}\\left(x\\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": "=2\\cdot\\:\\int\\:\\frac{1}{\\sin^{2}\\left(x\\right)}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{\\sin^{2}\\left(x\\right)}dx=-\\cot\\left(x\\right)$$", "result": "=2\\left(-\\cot\\left(x\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-2\\cot\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}-2\\cot\\left(x\\right)" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{x^{2}}{2}-2\\cot\\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", "practiceLink": "/practice/integration-practice#area=main&subtopic=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{x^{2}}{2}-2\\cot(x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }