What is the integral of cos^8(3x)sin(3x) ?

The integral of cos^8(3x)sin(3x) is -1/27 cos^9(3x)+C

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integral of cos^8(3x)sin(3x)

\int cos^{8} (3 x) sin (3 x) d x

- \frac{1}{27} cos^{9} (3 x) + C

Solution steps

\int cos^{8} (3 x) sin (3 x) d x

Apply u-substitution

: \int cos^{8} (u) sin (u) \frac{1}{3} d u

= \int cos^{8} (u) sin (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int cos^{8} (u) sin (u) d u

Apply u-substitution

: \int - v^{8} d v

= \frac{1}{3} \cdot \int - v^{8} d v

Take the constant out:

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

= \frac{1}{3} (- \int v^{8} d v)

Apply the Power Rule

: \frac{v ^{9}}{9}

= \frac{1}{3} (- \frac{v ^{9}}{9})

Substitute back

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

Simplify

\frac{1}{3} (- \frac{cos ^{9} ( 3 x )}{9}) : - \frac{1}{27} cos^{9} (3 x)

= - \frac{1}{27} cos^{9} (3 x)

Add a constant to the solution

= - \frac{1}{27} cos^{9} (3 x) + C

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

\int e^{\frac{- x}{5}} d x

\int \frac{6 x + 5}{x + 2} d x

\int x (2 x + 1)^{5} d x

What is the integral of cos^8(3x)sin(3x) ?
The integral of cos^8(3x)sin(3x) is -1/27 cos^9(3x)+C