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

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

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

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

- x + tan (2 x) + cos (2 x) + sin (2 x) tan (2 x) + C

Solution steps

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

Expand

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

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

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

\int 2 tan (2 x) sec (2 x) d x = cos (2 x) + sin (2 x) tan (2 x)

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

= - x + \frac{1}{2} tan (2 x) + cos (2 x) + sin (2 x) tan (2 x) + \frac{1}{2} tan (2 x)

Simplify

- x + \frac{1}{2} tan (2 x) + cos (2 x) + sin (2 x) tan (2 x) + \frac{1}{2} tan (2 x) : - x + tan (2 x) + cos (2 x) + sin (2 x) tan (2 x)

= - x + tan (2 x) + cos (2 x) + sin (2 x) tan (2 x)

Add a constant to the solution

= - x + tan (2 x) + cos (2 x) + sin (2 x) tan (2 x) + C

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