What is the integral of cos(θ)sqrt(sin(θ)+1) ?

The integral of cos(θ)sqrt(sin(θ)+1) is sin(θ)sqrt(sin(θ)+1)-1/3 (sin(θ)+1)^{3/2}+sqrt(sin(θ)+1)+C

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integral of cos(θ)sqrt(sin(θ)+1)

\int cos (θ) sin (θ) + 1 d θ

sin (θ) sin (θ) + 1 - \frac{1}{3} (sin (θ) + 1)^{\frac{3}{2}} + sin (θ) + 1 + C

Solution steps

\int cos (θ) sin (θ) + 1 d θ

Apply Integration By Parts

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

\int \frac{cos ( θ ) sin ( θ )}{2 sin ( θ ) + 1} d θ = \frac{1}{3} (sin (θ) + 1)^{\frac{3}{2}} - sin (θ) + 1

= sin (θ) sin (θ) + 1 - (\frac{1}{3} (sin (θ) + 1)^{\frac{3}{2}} - sin (θ) + 1)

Simplify

= sin (θ) sin (θ) + 1 - \frac{1}{3} (sin (θ) + 1)^{\frac{3}{2}} + sin (θ) + 1

Add a constant to the solution

= sin (θ) sin (θ) + 1 - \frac{1}{3} (sin (θ) + 1)^{\frac{3}{2}} + sin (θ) + 1 + C

What is the integral of cos(θ)sqrt(sin(θ)+1) ?
The integral of cos(θ)sqrt(sin(θ)+1) is sin(θ)sqrt(sin(θ)+1)-1/3 (sin(θ)+1)^{3/2}+sqrt(sin(θ)+1)+C