What is the integral of-cos(u)+1/3 cos^3(u) ?

The integral of-cos(u)+1/3 cos^3(u) is -2/3 sin(u)-1/9 sin^3(u)+C

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integral of-cos(u)+1/3 cos^3(u)

\int - cos (u) + \frac{1}{3} cos^{3} (u) d u

- \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u) + C

Solution steps

\int - cos (u) + \frac{1}{3} cos^{3} (u) d u

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 (u) d u + \int \frac{1}{3} cos^{3} (u) d u

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

\int \frac{1}{3} cos^{3} (u) d u = \frac{1}{3} (sin (u) - \frac{sin ^{3} ( u )}{3})

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

Simplify

- sin (u) + \frac{1}{3} (sin (u) - \frac{sin ^{3} ( u )}{3}) : - \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)

= - \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)

Add a constant to the solution

= - \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u) + C

What is the integral of-cos(u)+1/3 cos^3(u) ?
The integral of-cos(u)+1/3 cos^3(u) is -2/3 sin(u)-1/9 sin^3(u)+C