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

integral of cos^3(x)

The integral of

\cos^3(x)

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

.

Solution. We want to determine the integral of

\cos^3(x)

, i.e.:

\begin{align*} \int \cos^3(x) dx. \end{align*}

We saw here that

\cos^2(x) = 1 - \sin^2(x)

. So we get:

\begin{align*} \int \cos^3(x) dx = \int \cos^2(x)\cos(x) dx = \int (1 - \sin^2(x))\cos(x) dx. \end{align*}

We will apply the substitution method where

u = \sin(x)

. We saw here that

dU = \cos(x)dx

. Therefore, we get the following:

\begin{align*} \int (1 - \sin^2(x))\cos(x) dx &= \int (1 - u^2) du \\ &= u - \frac{1}{3}u^3 + C \\ &= \sin(x) - \frac{1}{3} \sin^3(x) + C. \end{align*}

Therefore, the integral of

\cos^3(x)

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