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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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