Question

What is the antiderivative of `x sin(x)`?

Answer 1

`intxsinxdx=-xcosx+sinx+C`

Explanation:

For this integral, we'll use integration by parts.

Choose your `u` to be `x`, so that way `(du)/dx=1->du=dx`. That means `dv=sinxdx->intdv=intsinxdx->v=-cosx`.

The integration by parts formula is:
`intudv=uv-intvdu`

We have `u=x`, `du=dx`, and `v=-cosx`. Substituting into the formula gives:
`intxsinxdx=-xcosx-int(-cosx)dx`
`color(white)(XX)=-xcosx+intcosxdx`
`color(white)(XX)=-xcosx+sinx+C`