Integrating By Parts

Integrating by parts is a useful technique for integrating expressions that appear to have no integral. The general form of integrating by parts is

UdV = UV - VdU

Note, our dU and dV are in terms of x.

What this amounts to is that our original function f(x) can be broken up into two parts; the first is a differentiable part "U", the second is an antidifferentiable part dv.

An example of such a function is.

f(x) = xe^x

The normal tools for integrating do not work here. Therefore, we will integrate by parts.

First we wish to identify our two parts, U and V. We will choose U = x and dV = e^x. This will be explained after the example. Our first step is to figure out what U, dU, V, and dV are.

U = x    dU = 1
dV = e^x    V = e^x

Note, we chose U and dV, and then we solved for dU and V.

So, we know that

UdV = UV - VdU

Now let's substitute the values we found .

xe^xdx = xe^x - e^xdx

This simplifies to

xe^xdx = xe^x - e^x

So we have our answer. Now we can check it. If I differentiate both sides I get

xe^xdx = (e^x + xe^x) - e^x

xe^x = xe^x

Success.

So, now, why didn't we choose U to be e^x and dV to be x? Let's try that out and see.

U = e^x    dU = e^x
dV = x    V = x

so now

xe^x = e^xx - xe^x

This is more complicated than our original integral! So there is some art to integrating by parts.