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

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

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

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

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

Solution steps

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

Apply u-substitution

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

= \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 = sin^{2} (x) + π

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

Add a constant to the solution

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

f (x) = e^{s i n (x)}

y = 2 x^{3} - 8 x, (- 2, 0)

\int e^{- t (s + 1)} d t

\frac{e ^{- s}}{1 - s}

f (t) = 2^{t^{3}}

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