Problems in calculus whether its the AP Calculus exam or second semester college calculus often involve integration by parts where one function is a natural logarithm. In this post I’m going to solve a problem out of my calculus book:

Let’s remind ourselves how to do integration by parts:

At first glance, we might be tempted to take:

But if we did that, we would be faced with the problem of finding:

So we’ve hit a wall. Instead, we will approach the problem by taking the natural logarithm to be u, and then we’ve got:

So the integral becomes:

Evaluating the integral we get the final result:

This example illustrates that the right choice of u and dv can save a bit of time and effort when solving integration by parts problems in calculus.

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