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

The integral of (tan(x)+sec(x))^2 is -x+2tan(x)+2sec(x)+C

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integral of (tan(x)+sec(x))^2

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

- x + 2 tan (x) + 2 sec (x) + C

Solution steps

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

Expand

(tan (x) + sec (x))^{2} : tan^{2} (x) + 2 tan (x) sec (x) + sec^{2} (x)

= \int tan^{2} (x) + 2 tan (x) sec (x) + sec^{2} (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 tan^{2} (x) d x + \int 2 tan (x) sec (x) d x + \int sec^{2} (x) d x

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

\int 2 tan (x) sec (x) d x = 2 sec (x)

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

= - x + tan (x) + 2 sec (x) + tan (x)

Simplify

= - x + 2 tan (x) + 2 sec (x)

Add a constant to the solution

= - x + 2 tan (x) + 2 sec (x) + C

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