# Find the antiderivative of tan2 (x) dx

We need to find the antiderivative of tan2x

### Solution

We know that tan x can be expressed in sin and cos as

tan x = sin / cos x

Hence

$\tan ^{2}x = \sin ^{2}x / \cos ^{2}x$ $\int tan ^{2}x. dx = \int sin ^{2}x / cos ^{2}x . dx$—————-(i)

We know from the trigonometric identity that

sin 2x + cos2x = 1

or sin2x= 1 – cos2x

Substituting sin2x= 1 – cos2x in equation (i) we get

= $\int 1 -\cos ^{2}x / \cos ^{2}x . dx$

= $\int 1 / \cos ^{2}x – \cos ^{2}x. dx$

= $\int 1 / \cos ^{2}x.dx – \int 1.dx$

=$\tan x – x + c$

Antiderivative of tan2x= $\tan x – x + c$