What is the integral of sin^5(3x)cos(3x) ?

The integral of sin^5(3x)cos(3x) is 1/18 sin^6(3x)+C

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integral of sin^5(3x)cos(3x)

\int sin^{5} (3 x) cos (3 x) d x

\frac{1}{18} sin^{6} (3 x) + C

Solution steps

\int sin^{5} (3 x) cos (3 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}{15} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = sin^{5} (3 x)

= \frac{1}{15} \cdot \frac{5}{6} (sin^{5} (3 x))^{\frac{6}{5}}

Simplify

\frac{1}{15} \cdot \frac{5}{6} (sin^{5} (3 x))^{\frac{6}{5}} : \frac{1}{18} sin^{6} (3 x)

= \frac{1}{18} sin^{6} (3 x)

Add a constant to the solution

= \frac{1}{18} sin^{6} (3 x) + C

f (x) = x^{2} - 5, (- 4, 11)

\int_{0}^{10} 1000 t e^{- 0.5 t} d t

\frac{d}{d θ} (e^{i θ})

\int 2 cos^{2} (θ) - \frac{1}{2} d θ

\frac{\partial}{\partial y} ((x^{2} - 2 x) cos (y))

What is the integral of sin^5(3x)cos(3x) ?
The integral of sin^5(3x)cos(3x) is 1/18 sin^6(3x)+C