What is the integral of tan^2(5x) ?

The integral of tan^2(5x) is -x+1/5 tan(5x)+C

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integral of tan^2(5x)

\int tan^{2} (5 x) d x

- x + \frac{1}{5} tan (5 x) + C

Solution steps

\int tan^{2} (5 x) d x

Rewrite using trig identities

= \int - 1 + sec^{2} (5 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d x + \int sec^{2} (5 x) d x

\int 1 d x = x

\int sec^{2} (5 x) d x = \frac{1}{5} tan (5 x)

= - x + \frac{1}{5} tan (5 x)

Add a constant to the solution

= - x + \frac{1}{5} tan (5 x) + C

What is the integral of tan^2(5x) ?
The integral of tan^2(5x) is -x+1/5 tan(5x)+C