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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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