What is the integral of sin^{12}(x)cos^3(x) ?

The integral of sin^{12}(x)cos^3(x) is (sin^{13}(x))/(13)-(sin^{15}(x))/(15)+C

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integral of sin^{12}(x)cos^3(x)

\int sin^{12} (x) cos^{3} (x) d x

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

Solution steps

\int sin^{12} (x) cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - sin^{2} (x)) cos (x) sin^{12} (x) d x

Apply u-substitution

: \int u^{12} (1 - u^{2}) d u

= \int u^{12} (1 - u^{2}) d u

Expand

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

= \int 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^{12} d u - \int u^{14} d u

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

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

= \frac{u ^{13}}{13} - \frac{u ^{15}}{15}

Substitute back

u = sin (x)

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

Add a constant to the solution

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

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What is the integral of sin^{12}(x)cos^3(x) ?
The integral of sin^{12}(x)cos^3(x) is (sin^{13}(x))/(13)-(sin^{15}(x))/(15)+C