What is the integral of cos^3(θ)sqrt(sin(θ)) ?

The integral of cos^3(θ)sqrt(sin(θ)) is 2/3 sin^{3/2}(θ)-2/7 sin^{7/2}(θ)+C

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integral of cos^3(θ)sqrt(sin(θ))

\int cos^{3} (θ) sin (θ) d θ

\frac{2}{3} sin^{\frac{3}{2}} (θ) - \frac{2}{7} sin^{\frac{7}{2}} (θ) + C

Solution steps

\int cos^{3} (θ) sin (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

u (1 - u^{2}) : u - u^{\frac{5}{2}}

= \int u - u^{\frac{5}{2}} 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 d u - \int u^{\frac{5}{2}} d u

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= \frac{2}{3} u^{\frac{3}{2}} - \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = sin (θ)

= \frac{2}{3} sin^{\frac{3}{2}} (θ) - \frac{2}{7} sin^{\frac{7}{2}} (θ)

Add a constant to the solution

= \frac{2}{3} sin^{\frac{3}{2}} (θ) - \frac{2}{7} sin^{\frac{7}{2}} (θ) + C

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What is the integral of cos^3(θ)sqrt(sin(θ)) ?
The integral of cos^3(θ)sqrt(sin(θ)) is 2/3 sin^{3/2}(θ)-2/7 sin^{7/2}(θ)+C