What is the integral of cos^3(x)-1 ?

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

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integral of cos^3(x)-1

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

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

Solution steps

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

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

\int 1 d x = x

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

Add a constant to the solution

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

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