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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \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 = cos (x)

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

Add a constant to the solution

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

\int \frac{1 - x ^{2}}{x ^{2}} d x

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