What is u-Substitution and How Do I Use It in Integration?

[chez_butt23]

Homework Statement

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:
\int4x\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.

 

[Mark44]

Homework Statement

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:
\int4x\sqrt{x^{2}+1}dx

It says that u=x^2+1 and that du/2=xdx.

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

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.

 

[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?

 

[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.

 

[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?

 

[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...

 

[SammyS]

...
I have this example from my textbook:
\int4x\sqrt{x^{2}+1}dx

It says that u=x^2+1 and that du/2=xdx. Where did they get this from? ...

If I take 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?

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

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

Now, divide by 2.

 

[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.

 

[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?

 

[e^(i Pi)+1=0]

yup.

 

[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?