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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int e^{\frac{1}{x}} \frac{1}{x ^{2}} d x

\int \frac{1}{x + 8} d x

\int \frac{4}{cos ( x )} d x

\int (1 - 3 x^{2}) d x

\int \frac{1}{1 - 2 x + 2 x ^{2}} d x

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