What is the integral of tan^8(x)sec^6(x) ?

The integral of tan^8(x)sec^6(x) is (tan^9(x))/9+(2tan^{11}(x))/(11)+(tan^{13}(x))/(13)+C

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integral of tan^8(x)sec^6(x)

\int tan^{8} (x) sec^{6} (x) d x

\frac{tan ^{9} ( x )}{9} + \frac{2 tan ^{11} ( x )}{11} + \frac{tan ^{13} ( x )}{13} + C

Solution steps

\int tan^{8} (x) sec^{6} (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int u^{8} (1 + u^{2})^{2} d u

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

Expand

u^{8} (1 + u^{2})^{2} : u^{8} + 2 u^{10} + u^{12}

= \int u^{8} + 2 u^{10} + u^{12} 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^{8} d u + \int 2 u^{10} d u + \int u^{12} d u

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

\int 2 u^{10} d u = \frac{2 u ^{11}}{11}

\int u^{12} d u = \frac{u ^{13}}{13}

= \frac{u ^{9}}{9} + \frac{2 u ^{11}}{11} + \frac{u ^{13}}{13}

Substitute back

u = tan (x)

= \frac{tan ^{9} ( x )}{9} + \frac{2 tan ^{11} ( x )}{11} + \frac{tan ^{13} ( x )}{13}

Add a constant to the solution

= \frac{tan ^{9} ( x )}{9} + \frac{2 tan ^{11} ( x )}{11} + \frac{tan ^{13} ( x )}{13} + C

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What is the integral of tan^8(x)sec^6(x) ?
The integral of tan^8(x)sec^6(x) is (tan^9(x))/9+(2tan^{11}(x))/(11)+(tan^{13}(x))/(13)+C