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

The integral of cos^3(3x)sin(3x) is -1/12 cos^4(3x)+C

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

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

- \frac{1}{12} cos^{4} (3 x) + C

Solution steps

\int cos^{3} (3 x) sin (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{3}{4} u^{\frac{4}{3}}

= - \frac{1}{9} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = cos^{3} (3 x)

= - \frac{1}{9} \cdot \frac{3}{4} (cos^{3} (3 x))^{\frac{4}{3}}

Simplify

- \frac{1}{9} \cdot \frac{3}{4} (cos^{3} (3 x))^{\frac{4}{3}} : - \frac{1}{12} cos^{4} (3 x)

= - \frac{1}{12} cos^{4} (3 x)

Add a constant to the solution

= - \frac{1}{12} cos^{4} (3 x) + C

x^{- \frac{5}{4}}

\int \frac{1}{x 9 x ^{2} - 9} d x

\int x^{3} 1 + 36 x^{2} d x

\int (e^{- \frac{x}{5}} + 5 x^{4}) d x

h (t) = (t + 4)^{\frac{2}{3}} (2 t^{2} - 4)^{3}

What is the integral of cos^3(3x)sin(3x) ?
The integral of cos^3(3x)sin(3x) is -1/12 cos^4(3x)+C