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

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

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

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

Solution steps

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

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

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

\int \frac{1}{x} 25 - x^{2} d x

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