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

The integral of (sec(x))^2(tan(x))^2 is (tan^3(x))/3+C

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

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

\frac{tan ^{3} ( x )}{3} + C

Solution steps

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

Simplify

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

Rewrite using trig identities

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

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{u ^{3}}{3}

Substitute back

u = tan (x)

= \frac{tan ^{3} ( x )}{3}

Add a constant to the solution

= \frac{tan ^{3} ( x )}{3} + C

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What is the integral of (sec(x))^2(tan(x))^2 ?
The integral of (sec(x))^2(tan(x))^2 is (tan^3(x))/3+C