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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{3 u}{6} d u

= \int - \frac{3 u}{6} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int 3 u d u

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 = cos^{3} (2 x)

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

Simplify

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

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

Add a constant to the solution

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

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