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

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

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

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

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

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos^{3} (3 x) d x - \int cos (3 x) d x

\int cos^{3} (3 x) d x = \frac{1}{3} (sin (3 x) - \frac{sin ^{3} ( 3 x )}{3})

\int cos (3 x) d x = \frac{1}{3} sin (3 x)

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

Simplify

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

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

Add a constant to the solution

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

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