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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + cos (x)

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

Add a constant to the solution

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

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

\frac{( 51 ln ( 8 ) - 51 )}{( ln ( 8 ) ) ^{2}}

\int x e^{(4 - x^{2})} d x

\int 34 x d x

\frac{\partial}{\partial x} (7 e^{x^{5} y})

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