integral of tan(x)

{ "query": { "display": "integral $$\\tan\\left(x\\right)$$", "symbolab_question": "PRE_CALC#integral \\tan(x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\ln\\left|\\cos(x)\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\tan\\left(x\\right)dx=-\\ln\\left|\\cos\\left(x\\right)\\right|+C$$", "input": "\\int\\:\\tan\\left(x\\right)dx", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\tan\\left(x\\right)dx", "result": "=\\int\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}dx", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\tan\\left(x\\right)=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=\\int\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}dx" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79RRuELv6RkEHtJl8t4ckN3RlZT3gzEvL+gTfmAEjQlFb4uwyGI2kmgF/HKAxbWqIYttiiWeV9UGDWszuBKZIYKs0Dp/9aGvLyBh3MeZ+oBsFK1jpeLiOzCwg/IFPtzinKhasWDrVeP81P1uRaV5406BBTEk/JQ2cZ9WKuRzClU7DcFgmo2GmeG2Kh9EaSPQP8xQp8yk2EdviTvfhz6ruO+9RQa+VFzMcO8uN2y3ZuWk" } }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\sin\\left(x\\right)}{\\cos\\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: $$u=\\cos\\left(x\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-\\sin\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$", "result": "=-\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoTIPsH/5VFEfonU6bvi80j8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zA92cpyjnQxeYFWLLJRXAqw02ZR5clxTmOwI/5g0CzzvDtz8RMf2ztf85Qhda6goD78yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-\\sin\\left(x\\right)dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{\\sin\\left(x\\right)}{u}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{\\sin\\left(x\\right)}{u}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right):{\\quad}-\\frac{1}{u}$$", "input": "\\frac{\\sin\\left(x\\right)}{u}\\left(-\\frac{1}{\\sin\\left(x\\right)}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{\\sin\\left(x\\right)}{u}\\cdot\\:\\frac{1}{\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{\\sin\\left(x\\right)\\cdot\\:1}{u\\sin\\left(x\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sin\\left(x\\right)$$", "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+4/5L7s7xStY6Xi4jswsIPyBT7c4pzSeYeCCLn7ZiGJmsXJgwkjK576WENGsoCX8pronHaqWSlQj8vi8h0NbzZABdtckVuxnvL7f2wjac+8KPowiAvKpT4Z8AS0JWpepKyA47Xdix42dP1ViqQ+THQU3fszrZTmw1q+hjFEjKzJRgYLTRGST/eN5UZxpetCGvRiIc8zCXMkMik0z8GIqYP+o/57Wi0=" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{u}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": "=-\\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 $$u=\\cos\\left(x\\right)$$", "result": "=-\\ln\\left|\\cos\\left(x\\right)\\right|" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\ln\\left|\\cos\\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", "practiceTopic": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\ln\\left|\\cos(x)\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }