What is the integral of sqrt(cos(θ))sin(2θ) ?

The integral of sqrt(cos(θ))sin(2θ) is -4/5 cos^{5/2}(θ)+C

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integral of sqrt(cos(θ))sin(2θ)

\int cos (θ) sin (2 θ) d θ

- \frac{4}{5} cos^{\frac{5}{2}} (θ) + C

Solution steps

\int cos (θ) sin (2 θ) d θ

Rewrite using trig identities

= \int cos (θ) \cdot 2 cos (θ) sin (θ) d θ

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos (θ) cos (θ) sin (θ) d θ

Apply Integration By Parts

: - \frac{2}{3} cos^{\frac{5}{2}} (θ) - \int \frac{2}{3} sin (θ) cos^{\frac{3}{2}} (θ) d θ

= 2 (- \frac{2}{3} cos^{\frac{5}{2}} (θ) - \int \frac{2}{3} sin (θ) cos^{\frac{3}{2}} (θ) d θ)

\int \frac{2}{3} sin (θ) cos^{\frac{3}{2}} (θ) d θ = - \frac{4}{15} cos^{\frac{5}{2}} (θ)

= 2 (- \frac{2}{3} cos^{\frac{5}{2}} (θ) - (- \frac{4}{15} cos^{\frac{5}{2}} (θ)))

Simplify

2 (- \frac{2}{3} cos^{\frac{5}{2}} (θ) - (- \frac{4}{15} cos^{\frac{5}{2}} (θ))) : - \frac{4}{5} cos^{\frac{5}{2}} (θ)

= - \frac{4}{5} cos^{\frac{5}{2}} (θ)

Add a constant to the solution

= - \frac{4}{5} cos^{\frac{5}{2}} (θ) + C

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What is the integral of sqrt(cos(θ))sin(2θ) ?
The integral of sqrt(cos(θ))sin(2θ) is -4/5 cos^{5/2}(θ)+C