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

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

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integral of (cos(x))/(sqrt(sin^3(x)))

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

- \frac{2}{sin ( x )} + C

Solution steps

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

sin^{3} (x) = sin^{\frac{3}{2}} (x),

assuming

sin (x) \geq 0

= \int \frac{cos ( x )}{sin ^{\frac{3}{2}} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{\frac{3}{2}}} d u

= \int \frac{1}{u ^{\frac{3}{2}}} d u

Apply the Power Rule

: - \frac{2}{u}

= - \frac{2}{u}

Substitute back

u = sin (x)

= - \frac{2}{sin ( x )}

Add a constant to the solution

= - \frac{2}{sin ( x )} + C

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