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

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

integral of 1-cos^3(x)

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

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

Solution steps

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

\int 1 d x = x

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

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

Simplify

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

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

Add a constant to the solution

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

\int cos (4 x) cos (- 3 x) d x

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