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

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

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integral of cos(x)sqrt(sin(x)+1)

\int cos (x) sin (x) + 1 d x

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

Solution steps

\int cos (x) sin (x) + 1 d x

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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