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

Question 1

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

Question 2

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

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

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

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

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