What is the integral of sin^3(2x)*cos(2x) ?

The integral of sin^3(2x)*cos(2x) is 1/8 sin^4(2x)+C

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integral of sin^3(2x)*cos(2x)

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

\frac{1}{8} sin^{4} (2 x) + C

Solution steps

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

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= \frac{1}{6} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = sin^{3} (2 x)

= \frac{1}{6} \cdot \frac{3}{4} (sin^{3} (2 x))^{\frac{4}{3}}

Simplify

\frac{1}{6} \cdot \frac{3}{4} (sin^{3} (2 x))^{\frac{4}{3}} : \frac{1}{8} sin^{4} (2 x)

= \frac{1}{8} sin^{4} (2 x)

Add a constant to the solution

= \frac{1}{8} sin^{4} (2 x) + C

\int_{1}^{\infty} \frac{4}{x - 1} d x

800 x - 0.1 x^{2}

n = 1 \sum \infty \frac{9 n}{4 n + 9}

\frac{d}{d x} (\frac{2}{1 - 3 x})

x \to 0 lim (\frac{x}{sin ( 3 x )})

What is the integral of sin^3(2x)*cos(2x) ?
The integral of sin^3(2x)*cos(2x) is 1/8 sin^4(2x)+C