Resolved: "Find the antiderivative of tan2(x)dx."

animeagan0o8

Answered question

2023-01-17

Find the antiderivative of

\tan^{2} (x) d x

cycloidey29

Beginner2023-01-18Added 5 answers

Compute the antiderivative:
The antiderivative can be found as,

\int \tan^{2} (x) d x = \int \frac{\sin^{2} (x)}{\cos^{2} (x)} d x

= \int \frac{(1 - \cos^{2} (x))}{\cos^{2} (x)} d x (∵ \sin^{2} (x) + \cos^{2} (x) = 1 \Rightarrow \sin^{2} (x) = 1 - \cos^{2} (x))

= \int (\frac{1}{\cos^{2} (x)} - \frac{\cos^{2} (x)}{\cos^{2} (x)}) d x

= \int ({s e c}^{2} (x) - 1) d x (∵ \frac{1}{\cos^{2} (x)} = {s e c}^{2} (x))

= \int {s e c}^{2} (x) d x - \int 1 d x

= \tan (x) - x + C (∵ \int {s e c}^{2} (x) d x = \tan (x), \int 1 d x = x)

Thus, the antiderivative of

\tan^{2} (x) d x

\tan (x) - x + C

, where

C

is constant of integration.

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