What is the integral of sin^{10}(θ)cos^3(θ) ?

The integral of sin^{10}(θ)cos^3(θ) is (sin^{11}(θ))/(11)-(sin^{13}(θ))/(13)+C

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integral of sin^{10}(θ)cos^3(θ)

\int sin^{10} (θ) cos^{3} (θ) d θ

\frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13} + C

Solution steps

\int sin^{10} (θ) cos^{3} (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

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

= \int 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^{10} d u - \int u^{12} d u

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

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

= \frac{u ^{11}}{11} - \frac{u ^{13}}{13}

Substitute back

u = sin (θ)

= \frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13}

Add a constant to the solution

= \frac{sin ^{11} ( θ )}{11} - \frac{sin ^{13} ( θ )}{13} + C

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What is the integral of sin^{10}(θ)cos^3(θ) ?
The integral of sin^{10}(θ)cos^3(θ) is (sin^{11}(θ))/(11)-(sin^{13}(θ))/(13)+C