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} (ln (sec^{3} (x^{4} - 1)))

x^{3}

n = 0 \sum \infty cos^{2} (n)

\frac{d}{d x} (\frac{x}{a} + \frac{a}{x})

\frac{\partial}{\partial x} (2 x^{\frac{1}{2}})

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