Integrating x√(x^2+a^2) using Substitution Method

[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.

 

[radou]

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

It's correct, if a is a constant.

 

[arildno]

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

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

 

[Office_Shredder]

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

 

[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!

 

[arildno]

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

 

[sapiental]

ah ok, I understand now. thanks again.