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

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

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

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

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

Solution steps

\int sin^{5} (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{5}{6} u^{\frac{6}{5}}

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

Substitute back

u = sin^{5} (2 x)

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{2 x + 1}{x ^{2} + 2 x - 3} d x

\int e^{- 2 k x} d x

y = 7 + 9 x

\frac{d}{d x} (5 sec (x) - 10 cos (x))

(x^{2} + 3) y^{^{'}} = x y

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