What is the integral of sin^{10}(x)cos^5(x) ?

The integral of sin^{10}(x)cos^5(x) is (sin^{11}(x))/(11)-(2sin^{13}(x))/(13)+(sin^{15}(x))/(15)+C

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integral of sin^{10}(x)cos^5(x)

\int sin^{10} (x) cos^{5} (x) d x

\frac{sin ^{11} ( x )}{11} - \frac{2 sin ^{13} ( x )}{13} + \frac{sin ^{15} ( x )}{15} + C

Solution steps

\int sin^{10} (x) cos^{5} (x) d x

Rewrite using trig identities

= \int (1 - sin^{2} (x))^{2} cos (x) sin^{10} (x) d x

Apply u-substitution

: \int u^{10} (1 - u^{2})^{2} d u

= \int u^{10} (1 - u^{2})^{2} d u

Expand

u^{10} (1 - u^{2})^{2} : u^{10} - 2 u^{12} + u^{14}

= \int u^{10} - 2 u^{12} + u^{14} 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^{12} d u + \int u^{14} d u

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

\int 2 u^{12} d u = \frac{2 u ^{13}}{13}

\int u^{14} d u = \frac{u ^{15}}{15}

= \frac{u ^{11}}{11} - \frac{2 u ^{13}}{13} + \frac{u ^{15}}{15}

Substitute back

u = sin (x)

= \frac{sin ^{11} ( x )}{11} - \frac{2 sin ^{13} ( x )}{13} + \frac{sin ^{15} ( x )}{15}

Add a constant to the solution

= \frac{sin ^{11} ( x )}{11} - \frac{2 sin ^{13} ( x )}{13} + \frac{sin ^{15} ( x )}{15} + C

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What is the integral of sin^{10}(x)cos^5(x) ?
The integral of sin^{10}(x)cos^5(x) is (sin^{11}(x))/(11)-(2sin^{13}(x))/(13)+(sin^{15}(x))/(15)+C