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

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

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integral of (tan(θ)+sec(θ))^2

\int (tan (θ) + sec (θ))^{2} d θ

- θ + 2 tan (θ) + 2 sec (θ) + C

Solution steps

\int (tan (θ) + sec (θ))^{2} d θ

Expand

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

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

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

\int tan^{2} (θ) d θ = - θ + tan (θ)

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

\int sec^{2} (θ) d θ = tan (θ)

= - θ + tan (θ) + 2 sec (θ) + tan (θ)

Simplify

= - θ + 2 tan (θ) + 2 sec (θ)

Add a constant to the solution

= - θ + 2 tan (θ) + 2 sec (θ) + C

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