Use integration by parts to find

taking *u* = sin *x* for the first integration.

Answer

If

*u* = sin *x*

*v'* = *e*^{x}

then

*u' *= cos* x*

*v* = *e*^{x}

When we stuff everything into the integration-by-parts formula, we get

Now we need to use integration by parts again to find

Remembering the lessons of the previous example, we'll keep *u* and *v*' with their same parts by taking

*u* = cos *x*

*v'* = *e*^{x}

Then

*u'* = -sin *x*

*v* = *e*^{x}

We put this into the formula and get

Putting this back into the first application of the formula for gives us

(being careful to properly distribute the negative sign!).

Again, we've found a formula we can solve for .