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

Take the constant out:

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

= - \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}})

Simplify

= \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