What is the integral of cos(2t)sin^5(2t) ?

The integral of cos(2t)sin^5(2t) is 1/12 sin^6(2t)+C

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integral of cos(2t)sin^5(2t)

\int cos (2 t) sin^{5} (2 t) d t

\frac{1}{12} sin^{6} (2 t) + C

Solution steps

\int cos (2 t) sin^{5} (2 t) d t

Apply u-substitution

: \int \frac{5 u}{10} d u

= \int \frac{5 u}{10} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int 5 u d u

Apply the Power Rule

: \frac{5}{6} u^{\frac{6}{5}}

= \frac{1}{10} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = sin^{5} (2 t)

= \frac{1}{10} \cdot \frac{5}{6} (sin^{5} (2 t))^{\frac{6}{5}}

Simplify

\frac{1}{10} \cdot \frac{5}{6} (sin^{5} (2 t))^{\frac{6}{5}} : \frac{1}{12} sin^{6} (2 t)

= \frac{1}{12} sin^{6} (2 t)

Add a constant to the solution

= \frac{1}{12} sin^{6} (2 t) + C

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What is the integral of cos(2t)sin^5(2t) ?
The integral of cos(2t)sin^5(2t) is 1/12 sin^6(2t)+C