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

The integral of (tan^2(θ))/(sec(θ)) is -sin(θ)+ln|tan(θ)+sec(θ)|+C

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integral of (tan^2(θ))/(sec(θ))

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

- sin (θ) + ln ∣ tan (θ) + sec (θ) ∣ + C

Solution steps

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

Rewrite using trig identities

= \int \frac{- 1 + sec ^{2} ( θ )}{sec ( θ )} d θ

Expand

\frac{- 1 + sec ^{2} ( θ )}{sec ( θ )} : - \frac{1}{sec ( θ )} + sec (θ)

= \int - \frac{1}{sec ( θ )} + sec (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 \frac{1}{sec ( θ )} d θ + \int sec (θ) d θ

\int \frac{1}{sec ( θ )} d θ = sin (θ)

\int sec (θ) d θ = ln ∣ tan (θ) + sec (θ) ∣

= - sin (θ) + ln ∣ tan (θ) + sec (θ) ∣

Add a constant to the solution

= - sin (θ) + ln ∣ tan (θ) + sec (θ) ∣ + C

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