What is the integral of tan^5(x)sec^7(x) ?

The integral of tan^5(x)sec^7(x) is (sec^{11}(x)}{11}-\frac{2sec^9(x))/9+(sec^7(x))/7+C

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integral of tan^5(x)sec^7(x)

\int tan^{5} (x) sec^{7} (x) d x

\frac{sec ^{11} ( x )}{11} - \frac{2 sec ^{9} ( x )}{9} + \frac{sec ^{7} ( x )}{7} + C

Solution steps

\int tan^{5} (x) sec^{7} (x) d x

Rewrite using trig identities

= \int (- 1 + sec^{2} (x))^{2} tan (x) sec^{7} (x) d x

Apply u-substitution

: \int u^{6} (u^{2} - 1)^{2} d u

= \int u^{6} (u^{2} - 1)^{2} d u

Expand

u^{6} (u^{2} - 1)^{2} : u^{10} - 2 u^{8} + u^{6}

= \int u^{10} - 2 u^{8} + u^{6} 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^{10} d u - \int 2 u^{8} d u + \int u^{6} d u

\int u^{10} d u = \frac{u ^{11}}{11}

\int 2 u^{8} d u = \frac{2 u ^{9}}{9}

\int u^{6} d u = \frac{u ^{7}}{7}

= \frac{u ^{11}}{11} - \frac{2 u ^{9}}{9} + \frac{u ^{7}}{7}

Substitute back

u = sec (x)

= \frac{sec ^{11} ( x )}{11} - \frac{2 sec ^{9} ( x )}{9} + \frac{sec ^{7} ( x )}{7}

Add a constant to the solution

= \frac{sec ^{11} ( x )}{11} - \frac{2 sec ^{9} ( x )}{9} + \frac{sec ^{7} ( x )}{7} + C

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What is the integral of tan^5(x)sec^7(x) ?
The integral of tan^5(x)sec^7(x) is (sec^{11}(x)}{11}-\frac{2sec^9(x))/9+(sec^7(x))/7+C