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

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

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

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

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

Solution steps

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

\int cos (x) d x = sin (x)

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

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

Simplify

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

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

Add a constant to the solution

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

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