What is the integral of (tan(x))^5(sec(x))^2 ?

The integral of (tan(x))^5(sec(x))^2 is (tan^6(x))/6+C

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integral of (tan(x))^5(sec(x))^2

\int (tan (x))^{5} (sec (x))^{2} d x

\frac{tan ^{6} ( x )}{6} + C

Solution steps

\int (tan (x))^{5} (sec (x))^{2} d x

Simplify

= \int tan^{5} (x) sec^{2} (x) d x

Rewrite using trig identities

= \int (1 + tan^{2} (x)) tan^{5} (x) d x

Apply u-substitution

: \int u^{5} d u

= \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{u ^{6}}{6}

Substitute back

u = tan (x)

= \frac{tan ^{6} ( x )}{6}

Add a constant to the solution

= \frac{tan ^{6} ( x )}{6} + C

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What is the integral of (tan(x))^5(sec(x))^2 ?
The integral of (tan(x))^5(sec(x))^2 is (tan^6(x))/6+C