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

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

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

integral

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

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

Solution steps

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

Rewrite using trig identities

= \int \frac{( 1 - sin ^{2} ( x ) ) cos ( x )}{sin ( x )} d x

Apply u-substitution

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

Expand

\frac{1 - u ^{2}}{u} : \frac{1}{u} - u^{\frac{3}{2}}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

Substitute back

u = sin (x)

= 2 sin (x) - \frac{2}{5} sin^{\frac{5}{2}} (x)

Add a constant to the solution

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

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