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

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

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integral of sqrt(sin(x)-sin^3(x))

\int sin (x) - sin^{3} (x) d x

\frac{2}{3} sin^{\frac{3}{2}} (x) + C

Solution steps

\int sin (x) - sin^{3} (x) d x

Simplify

sin (x) - sin^{3} (x) : cos (x) sin (x)

= \int cos (x) sin (x) d x

Apply u-substitution

: \int u d u

= \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 = sin (x)

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

Add a constant to the solution

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

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