What is the integral of (tan^5(x))/(sec^8(x)) ?

The integral of (tan^5(x))/(sec^8(x)) is -(cos^4(x))/4+(cos^6(x))/3-(cos^8(x))/8+C

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integral of (tan^5(x))/(sec^8(x))

\int \frac{tan ^{5} ( x )}{sec ^{8} ( x )} d x

- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

Solution steps

\int \frac{tan ^{5} ( x )}{sec ^{8} ( x )} d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{3} (x) d x

Apply u-substitution

: \int - u^{3} (1 - u^{2})^{2} d u

= \int - u^{3} (1 - u^{2})^{2} d u

Expand

- u^{3} (1 - u^{2})^{2} : - u^{3} + 2 u^{5} - u^{7}

= \int - u^{3} + 2 u^{5} - u^{7} 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 u^{3} d u + \int 2 u^{5} d u - \int u^{7} d u

\int u^{3} d u = \frac{u ^{4}}{4}

\int 2 u^{5} d u = \frac{u ^{6}}{3}

\int u^{7} d u = \frac{u ^{8}}{8}

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{3} - \frac{u ^{8}}{8}

Substitute back

u = cos (x)

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8}

Add a constant to the solution

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

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What is the integral of (tan^5(x))/(sec^8(x)) ?
The integral of (tan^5(x))/(sec^8(x)) is -(cos^4(x))/4+(cos^6(x))/3-(cos^8(x))/8+C