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

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

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

integral

cos^{3} (x) - cos (x)

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

Solution steps

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

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

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

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

Simplify

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

in t e g r a l sin (2 - 4 x)

in t e g r a l \frac{x}{( x ^{2} + a ^{2} )}

in t e g r a l \frac{1}{( x - 1 )}

in t e g r a l x^{3} (e^{x})^{2}

in t e g r a l \frac{1}{( x ^{3} + 1 )}

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