What is int x^2lnxdx?

1 Answer
Nov 9, 2015

Use f(x) = ln(x), g'(x) = x^2.
The result is: 1/3 x^3 ln(x) - 1/9 x^3 .

Explanation:

Use "integration by parts".

The rule is:
int f(x) g'(x) dx = f(x) g(x) - int f'(x) g(x) dx

A very important decision is now which part of your product is f(x) and which one is g'(x) since later, you will need to differentiate one of the terms and integrate the other one. With a wrong choice, you might never find a solution.

Here, the choice is not very hard: you really want to differentiate and not integrate ln(x) (trust me on that ;-) ), so choose f(x) = ln(x) and g'(x) = x^2.

Now, you need to compute f'(x) = 1/x and g(x) = 1/3 x^3.

At this point you are ready to apply the rule.

int x^2 ln(x) dx = ln(x) * 1/3 x^3 - int 1/x * 1/3 x^3 dx
= 1/3 x^3 ln(x) - 1/3 int x^2 dx
= 1/3 x^3 ln(x) - 1/3 * ( 1/3 x^3 )
= 1/3 x^3 ln(x) - 1/9 x^3
or, if you wish,
= 1/3 x^3 (ln(x) - 1/3)