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

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

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integral of (tan(x))^2

integral

(tan (x))^{2}

tan (x) - x + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

\int 1 d x = x

= tan (x) - x

Add a constant to the solution

= tan (x) - x + C