Home » Blog » Use integration by parts to evaluate the given integral

Use integration by parts to evaluate the given integral

by RoRi

Evaluate the following integral using integration by parts:

    \[ \int x \sin x \, dx. \]

To apply the integration by parts formula we let:

    \begin{align*} u &= x &\implies && du &= dx \\ dv &= \sin x \, dx&\implies && v &= -\cos x. \end{align*}

Then we use integration by parts

    \begin{align*} \int x \sin x \, dx &= \int u \, dv \\ &= uv - \int v \, du \\ &= -x \cos x + \int \cos x \, dx \\ &= \sin x - 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):