Help evaluating a definite integral using u-substitution

M

michael87

New member
Joined
Nov 29, 2012
Messages
3
Hello everyone, I was hoping someone could help me out here. I'm having a lot of trouble with this exercise in evaluating a definite integral.

The exercise is this: Evaluate the integral from 0 to 1 of " x^2(1+2x^2)^5 dx"

Using u-substitution these are the steps I took:

Let u = 1+2x^2
du/dx = 4x
1/4du = xdx

Now I rewrite the integral into the new form: 1/4 times the integral of " x(u)^5 du "

This is where I am completely lost. My u-substitution did not get rid of all the x variables. How do I get rid of that last x variable so that I can compute the anti-derivative and then evaluate the integral? I also tried using the substitution u = x^2 but that didn't work either.

Hope someone can help me out. Thanks.
 
Your first substitution was fine. u = 1 + 2x^2

You also have, since x > 0, \(\displaystyle x = \sqrt{\dfrac{u-1}{2}}\)
 
tkhunny said:
Your first substitution was fine. u = 1 + 2x^2

You also have, since x > 0, \(\displaystyle x = \sqrt{\dfrac{u-1}{2}}\)
Click to expand...

Okay, thanks, that makes sense. But now that gives me:

math_image.aspx


Which I could then rewrite, removing the constant multiple, as:

math_image.aspx


But now I'm stuck here (assuming I wrote everything correctly). Now my u-substitution hasn't resulted in a form which is easy to find the anti-derivative. So do I have to use another substitution? If so, how would that work? I don't understand why I'm so baffled by this problem, but I am.
 
michael87 said:
Okay, thanks, that makes sense. But now that gives me:

math_image.aspx


Which I could then rewrite, removing the constant multiple, as:

math_image.aspx


But now I'm stuck here (assuming I wrote everything correctly). Now my u-substitution hasn't resulted in a form which is easy to find the anti-derivative. So do I have to use another substitution? If so, how would that work? I don't understand why I'm so baffled by this problem, but I am.
Click to expand...

As you found out, applying 'u-substitution' directly - is of no help here.

I would attack this problem by expanding (1 + 2x2)5 - using binomial theorem - and integrating term by term.
 
Subhotosh Khan said:
As you found out, applying 'u-substitution' directly - is of no help here.

I would attack this problem by expanding (1 + 2x2)5 - using binomial theorem - and integrating term by term.
Click to expand...


Thanks, that's extremely frustrating! If there is any other way of approaching this integral other than multiplying everything out, I would love to hear it.

The answer I got was ridiculously long. The anti-derivative came to: (32/13)x^13 + (80/11)x^11 + (80/9)x^9 + (40/7)x^7 + 2x^5 + (1/3)x^3

Again, thanks so much for all the help!
 
michael87 said:
Thanks, that's extremely frustrating! If there is any other way of approaching this integral other than multiplying everything out, I would love to hear it.

The answer I got was ridiculously long. The anti-derivative came to: (32/13)x^13 + (80/11)x^11 + (80/9)x^9 + (40/7)x^7 + 2x^5 + (1/3)x^3

Again, thanks so much for all the help!
Click to expand...

That is not ridiculously long at all!

You can do this by trigonometric substitution also:

x = (1/√2) * tan(Θ)

But that is no bed of roses either.......
 
Agreed, and all, but it was worth the exposure to the idea of what to do when there is some extra 'x' sitting around after assembling du.
 
Last edited:
You must log in or register to reply here.
Top