What is the integral of (tan^7(x))/(sec^8(x)) ?

The integral of (tan^7(x))/(sec^8(x)) is (sin^8(x))/8+C

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integral of (tan^7(x))/(sec^8(x))

\int \frac{tan ^{7} ( x )}{sec ^{8} ( x )} d x

\frac{sin ^{8} ( x )}{8} + C

Solution steps

\int \frac{tan ^{7} ( x )}{sec ^{8} ( x )} d x

Rewrite using trig identities

= \int sin^{7} (x) cos (x) d x

Apply u-substitution

: \int w^{7} d w

= \int w^{7} d w

Apply the Power Rule

: \frac{w ^{8}}{8}

= \frac{w ^{8}}{8}

Substitute back

w = sin (x)

= \frac{sin ^{8} ( x )}{8}

Add a constant to the solution

= \frac{sin ^{8} ( x )}{8} + C

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What is the integral of (tan^7(x))/(sec^8(x)) ?
The integral of (tan^7(x))/(sec^8(x)) is (sin^8(x))/8+C