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

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

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integral of (sec(θ)-tan(θ))^2

\int (sec (θ) - tan (θ))^{2} d θ

- θ + 2 tan (θ) - 2 sec (θ) + C

Solution steps

\int (sec (θ) - tan (θ))^{2} d θ

Expand

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

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

\int sec^{2} (θ) d θ = tan (θ)

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

\int tan^{2} (θ) d θ = - θ + tan (θ)

= tan (θ) - 2 sec (θ) - 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

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