# Integration by substitution

1. Oct 22, 2006

### sapiental

Hello,

evaluate the following integral:

$$\int x \sqrt{x^2+a^2}dx$$

definite integral from 0 to a

what I did was

u = x^2 + a^2
du = 2xdx

1/2 sqrt(u)du

I just dropped the a^2 because we were finding the derivative of x but feel that it's very wrong.

Any suggestions are much appreciated.

thanks.

Last edited: Oct 22, 2006
2. Oct 22, 2006

It's correct, if a is a constant.

3. Oct 22, 2006

### arildno

Why do you feel it is wrong?
1. What is the derivative of a constant?

2. What are the corresponding u-limits compared to the given x-limits?

4. Oct 22, 2006

### Office_Shredder

Staff Emeritus
You'll notice that when you go from u to x again, the a2 term pops back in.

So it's not like you completely lost it

5. Oct 22, 2006

### sapiental

hey,

thanks again for all the help. I learn so much from this website and it helps me be much more confident with the problems.

d/dx of C = 0

and the corresponing u limits are 0 + a^2 and a^2 + a^2

what throws me off about dropping the a^2 is that a itself is one of the limits of integration.

thanks!

6. Oct 22, 2006

### arildno

Why should that matter??
Would you have accepted it if the limit was some number c instead?

7. Oct 22, 2006

### sapiental

ah ok, I understand now. thanks again.