Question

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

Answer 1

Answer:

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

Step-by-step explanation:

$\int {tan}^{3} x \: dx$

$\int \: tan \: x \times {tan}^{2} x \: dx$

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

distribute

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

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

$\int \: tan \: x \: {sec}^{2} x \: dx \: - \int \frac{sin \: x}{cos \: x} \: dx$

First integrand

let tan x = u

du = sec²x dx

Second integrand

let cos x = z

dz = -sin x dx

$= \int u \: du \: - \int - \frac{1}{z} dz$

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

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