# Issues with u-Substitution

1. Jul 2, 2012

### chez_butt23

1. The problem statement, all variables and given/known data
I am struggling with u-substitution. I understand that it essentially the undoing of the chain rule, but I don not get how to actually go about the procedure.

I have this example from my textbook:
$\int$4x$\sqrt{x^{2}+1}$dx

It says that u=x^2+1 and that du/2=xdx. Where did they get this from? How do I know what to use for u in any given equation. Once I have these to values (u and du) I know that it is just a matter of taking the integral andd then plugging x in for u. Any help is appreciated.

Last edited by a moderator: Jul 2, 2012
2. Jul 2, 2012

### Staff: Mentor

Do you know how to get the differential of u (i.e., du)?

3. Jul 2, 2012

### chez_butt23

If I tske the derivative of u, then that should give me du/dx=something. I then multiply by dx to get du. Is that what you mean?

4. Jul 2, 2012

### e^(i Pi)+1=0

That is right, you can treat it just like a fraction. Then isolate dx and substitute it into the original equation along with your U substitution. You'll lean what to substitute U for with experience. Basically, what ever you substitute U for, call it A for you'll end up with $\frac{1}{\frac{d}{dx} A}$. If you do it right, this factor will cancel with some other x in the integral, leaving you with an integral only in terms of U du.

Last edited: Jul 2, 2012
5. Jul 2, 2012

### chez_butt23

Thank you for the replies. I'm sorry, but I don't understand anything about A. I have never heard of this before. Could you elaborate please?

Also, are there any tricks to figuring out what u is?

6. Jul 2, 2012

### e^(i Pi)+1=0

That is the "trick". I said let A represent whatever you substitute for U. Then when you follow the method you'll end up with 1 over A prime. Will post an example, just a sec.....

7. Jul 2, 2012

### SammyS

Staff Emeritus
So, if $\displaystyle u=x^2+1\,,$ then $\displaystyle \frac{du}{dx}=2x\,.$

Therefore, $\displaystyle du=2x\,dx\,,$ correct ?

Now, divide by 2.

8. Jul 2, 2012

### e^(i Pi)+1=0

I forgot to write dx at the end of original function at top of page, I hope it's still clear.

Last edited: Jul 2, 2012
9. Jul 2, 2012

### chez_butt23

^That right there just solved my problems. Thank you so much for your help everyone I really appreciate it.

One more question though. How do I go about solving u-substitution problems for definite integrals? Is it the same but I plug in values of x and subtract at the end?

10. Jul 2, 2012

### e^(i Pi)+1=0

yup.

11. Jul 4, 2012

### chez_butt23

One more thing, are there any tricks to realizing if a problem is u-substitution or integration by parts? Or is it merely just a matter of doing one to see if it works?