Integration by Parts

To integrate a product of functions, we can integrate by parts. This is ineffect, an integrated form of the product rule used in differentiation.

For an integral of the form,

`int F(x)G(x)dx`

We can solve it by equating it to the standard form,

`int uv'dx = uv - int vu'dx`

By inspection we obtain the values of v and u, and their related forms,

`u=F(x)` and `v'=G(x)`

Therefore,

`u'=F(x)'` and `v=int G(x)dx`

We choose the particular combination of `u` and `v'`, such that their complementary functions, `u'` and `v` are simpler.

Example 1:

`int x exp(x)`

We choose,

`u=x` and `v'=exp(x)`

Then we find `u'` and `v`,

`u'=(du)/(dx)=(dx)/(dx)=1` and `v=int exp(x)dx=exp(x)`

We note that the derivative `u'` is equal one, which makes the second integral is simpler than the original.

Inserting these values of `v` and `u`` and their related functions, into the standard form we obtain,

`int x exp(x) = uv - int vu'dx`

`=int x exp(x)dx = xexp(x) - int exp(x) dx`

Therefore,

`int x exp(x)dx = xexp(x) - exp(x) = exp(x)(x-1)`