What is the integral of sin(x)(cos(x))^{3/2} ?

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

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integral of sin(x)(cos(x))^{3/2}

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

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

Solution steps

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

Simplify

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

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

Substitute back

u = cos (x)

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

Add a constant to the solution

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

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