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

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}}

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

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