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

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

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integral of (tan(u))^2

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

tan (u) - u + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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} ( u )} d u - \int 1 d u

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

\int 1 d u = u

= tan (u) - u

Add a constant to the solution

= tan (u) - u + C

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