# Thread: Integration by Substitution Question

1. ## Integration by Substitution Question

I have calculated this problem multiple times, but never obtain the same answer that Maple provides.

The integration limits are from 1 to 4, and the integrand is e raised to the square root of x, divided by the root of x.
I have substituted "u" for the upper square root of x, which leaves me with the new integral with an integrand of e raised to the "u", divided by 2, with integration limits from 1 to 2.

When I evaluate this, the answer I obtain is approximately 2.3354. Maple, however, provides me with an answer of approximately 9.3415.

This question has me banging my head against the wall , because I feel that I am right, but technology is telling me I am wrong!

If someone could tell me what I am doing wrong, or what Maple might be doing, that would be very much appreciated. I apologize for my lack of mathematical notation. If someone could please tell me how to do the symbols on here, that would be awesome!

2. Make the following substitution

to transform the integral into

.

Maple is correct.

3. I'll just elaborate a bit on what was shown above.
You didn't change the LIMITS of integration when making the substitution. Notice that "Prove It" manually included the "u =..." in the limits. If you wanted to avoid making this mistake ever again, you could explicitly write "x = " or "u = " for each limit of integration. But eventually, you'll get used to this kind of thing and will succeed without such helps.

4. Where does the 2 out front come from? If the upper root of x is u, its derivative will be 1/2*root of x. And thus, 1/root of x will be replaced by 1/2du, and the 1/2 can be pulled outside of the integral, leaving 1/2 multiplied by the integral of e^u, evaluated from 1 to 2, right? Where am I wrong in my thinking?
By the way, how are you all doing fancy notation in the forum?

5. Originally Posted by SC313
...
By the way, how are you all doing fancy notation in the forum?
Online LaTeX Equation Editor - create, integrate and download is how I do it.

The 2 out in front (in the first line) isn't the way most people would work this; it would appear later. If you're having trouble with getting the integrand to contain du (exactly), you might need to review u-substitution.

6. If u = Sqrt[x], then

du/dx = 1/(2Sqrt[x])

du = 1/(2Sqrt[x])dx

2du = 1/(Sqrt[x])dx.

So you don't replace 1/(Sqrt[x])dx with (1/2)du, you replace it with 2du.

7. I had everything right except it would be replaced by 2*du. I feel stupid! Sorry all!

8. Originally Posted by TheChaz
Online LaTeX Equation Editor - create, integrate and download is how I do it.

The 2 out in front (in the first line) isn't the way most people would work this; it would appear later. If you're having trouble with getting the integrand to contain du (exactly), you might need to review u-substitution.
Don't speak for most people...

9. Originally Posted by Prove It
Don't speak for most people...
The thing is... with a statement like "most people..." it can be quite burdensome to disProve It.

Maybe there's a way you could get the point across with using the imperative...

10. Thanks all! Do either of you know of a really good book which deals with applications of calculus, such as mixture problems?

11. Originally Posted by SC313
Thanks all! Do either of you know of a really good book which deals with applications of calculus, such as mixture problems?
I try to steer clear of any sort of application of math! Wish I could be more help...

12. Originally Posted by TheChaz
I try to steer clear of any sort of application of math! Wish I could be more help...
I hate it as well, but the teachers I have been graced with all adore applications. Well, what's a really good calculus book in general?

13. Originally Posted by SC313
I hate it as well, but the teachers I have been graced with all adore applications. Well, what's a really good calculus book in general?
Many forum users (here and elsewhere) like Amazon.com: Calculus, 4th edition (9780914098911): Michael Spivak: Books
(Spivak's Calculus). I personally like Amazon.com: A Tour of the Calculus (9780679747888): David Berlinski: Books , but I've heard more criticism about this book than praise!

Thanks!