## Example 7.2.6(a): Applying Integration by Parts

We need to identify two functions such thatIn our case we define

- we know the antiderivative of the first function - that function will be
g'.- the derivative of the second function is easier than the original function - that function will be
f.

*g'(x) = e*and

^{x}*f(x) = x*. Then we need to find

*G(x) = f(x) g(x)*, which in this case is

*G(x) = x e*.

^{x}Integration by parts now gives the answer:

wherex e^{x}dx = G(b) - G(a) - e^{x}dx =

= G(b) - G(a) - [ exp(b) - exp(a) ]

*G(x) = x e*.

^{x}