What is the integral of sin(x)sqrt((1+cos(x))^3) ?

The integral of sin(x)sqrt((1+cos(x))^3) is -2/5 (1+cos(x))^{5/2}+C

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integral of sin(x)sqrt((1+cos(x))^3)

\int sin (x) (1 + cos (x))^{3} d x

- \frac{2}{5} (1 + cos (x))^{\frac{5}{2}} + C

Solution steps

\int sin (x) (1 + cos (x))^{3} d x

Apply u-substitution

: \int - u^{\frac{3}{2}} d u

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= - \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 1 + cos (x)

= - \frac{2}{5} (1 + cos (x))^{\frac{5}{2}}

Add a constant to the solution

= - \frac{2}{5} (1 + cos (x))^{\frac{5}{2}} + C

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What is the integral of sin(x)sqrt((1+cos(x))^3) ?
The integral of sin(x)sqrt((1+cos(x))^3) is -2/5 (1+cos(x))^{5/2}+C