Home » Blog » Evaluate the integral using substitution

Evaluate the integral using substitution

Use the method of substitution to evaluate the following integral:

    \[ \int \sin^3 x \, dx. \]


First, we rearrange the integral to get something easier to work with,

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

Now, let

    \[ u = \cos x \quad \implies \quad du = - \sin x \, dx. \]

Then, we can evaluate the integral,

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

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):