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\int {tan}^(3) x \: dx
Evaluate the integral above​

User James
by
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1 Answer

4 votes

Answer:


\frac{ {tan}^(2) x}{2} + \ln( |cos \: x| ) + C

Explanation:


\int {tan}^(3) x \: dx


\int \: tan \: x * {tan}^(2) x \: dx


\int \: tan \: x( {sec}^(2) x - 1) \: dx

distribute


\int \: tan \: x \: {sec}^(2) x - tan \: x \: dx


\int \: tan \: x \: {sec}^(2) x \: dx \: - \int \: tan \: x \: dx


\int \: tan \: x \: {sec}^(2) x \: dx \: - \int (sin \: x)/(cos \: x) \: dx

First integrand

let tan x = u

du = sec²x dx

Second integrand

let cos x = z

dz = -sin x dx


= \int u \: du \: - \int - (1)/(z) dz


= \frac{ {u}^(2) }{2} + \ln( |z| ) + C


= \color{red}{ \boxed{ \frac{ {tan}^(2) x}{2} + \ln( |cos \: x| ) + C}}

User TwitchBronBron
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