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

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

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

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

=\(\tan x – x + c\)

Answer

Antiderivative of tan2x= \(\tan x – x + c\)

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