What is the integral of (1+tan^2(5x)) ?

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

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integral of (1+tan^2(5x))

\int (1 + tan^{2} (5 x)) d x

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

Solution steps

\int 1 + tan^{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 tan^{2} (5 x) d x

\int 1 d x = x

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

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

Simplify

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

Add a constant to the solution

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

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