# Thread: Definite integral with u substitution

1. ## Definite integral with u substitution

Integral 1 to 2 of (x^3)lnxdx

What do we use as u? I tried lnx and x^3 and it didnt seem to work.

2. ## Re: Definite integral with u substitution

The method is to integrate by parts.

3. ## Re: Definite integral with u substitution

You should remember the acronym LIATE: you want to choose for u whatever comes first in the list Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. In this case you would choose $u=\ln{x}$.

- Hollywood

4. ## Re: Definite integral with u substitution

Originally Posted by BobP
The method is to integrate by parts.
Yeah my friend and I realized this when we looked on the page before it said with product do integration by parts haha...Thanks!

5. ## Re: Definite integral with u substitution

Here's another method:

$\int_1^2 x^a dx=\frac{2^{1+a}-1}{1+a}$

Differentiate with respect to $a$:

$\int_1^2 x^a \log(x) dx=\frac{2^{1+a}\log(2)(1+a)-2^{1+a}+1}{(1+a)^2}$

Putting $a=3$, gives the answer

$\int_1^2 x^3 \log(x) dx= \frac{4\times 2^4 \log(2)-2^4+1}{4^2}=4\log(2)-\frac{15}{16}$