What is the integral of (tan(x))^8(sec(x))^2 ?

The integral of (tan(x))^8(sec(x))^2 is (tan^9(x))/9+C

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integral of (tan(x))^8(sec(x))^2

\int (tan (x))^{8} (sec (x))^{2} d x

\frac{tan ^{9} ( x )}{9} + C

Solution steps

\int (tan (x))^{8} (sec (x))^{2} d x

Simplify

= \int tan^{8} (x) sec^{2} (x) d x

Rewrite using trig identities

= \int (1 + tan^{2} (x)) tan^{8} (x) d x

Apply u-substitution

= \int u^{8} d u

Apply the Power Rule

= \frac{u ^{9}}{9}

Substitute back

u = tan (x)

= \frac{tan ^{9} ( x )}{9}

Add a constant to the solution

= \frac{tan ^{9} ( x )}{9} + C

What is the integral of (tan(x))^8(sec(x))^2 ?
The integral of (tan(x))^8(sec(x))^2 is (tan^9(x))/9+C