Question

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

Answer 1

`\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}}`