Homework Help: Definite integral involving base e and square root of variable as exponent

1. Jul 2, 2008

phyzmatix

1. The problem statement, all variables and given/known data

Calculate

$$\int_{0}^{4}\sqrt{x} e^{\sqrt{x})$$

3. The attempt at a solution

At first I tried substitution, but this didn't bring me anywhere since the integral is not of the form

$$\int f(g(x))g'(x)$$

My attempt at integration by parts also leads to an endless loop. I used

$$u = e^{\sqrt{x}}$$

Please give me some pointers here. I know I didn't include a whole attempt at a solution, but it's only to save myself some time with the typing (latex is still slow-going). However, if you really need me to include all I've done, I can do so.

2. Jul 2, 2008

gamesguru

Use substitution, u=Sqrt[x], du=1/(2Sqrt[x])dx so 2u du=dx.
Hope that helps.

3. Jul 2, 2008

CompuChip

I didn't work this out so I may be pointing you in the wrong direction. But what if you write the integrand as
$$\sqrt{x} e^{\sqrt{x}} \propto x \frac{\mathrm d}{\mathrm dx} e^{\sqrt{x}}$$

4. Jul 2, 2008

phyzmatix

Thanks gamesguru, but unless I'm doing something wrong, this doesn't seem to bring me any closer to the answer.

Hi CompuChip. Sorry, but I have no idea what this is that you've just done here Would you mind explaining?

EDIT: I just noticed that I forgot the "dx" in the OP. Sorry.

5. Jul 2, 2008

CompuChip

I've just rewritten the thing under the integral sign as a product of a (simple) function of x and a derivative w.r.t. x. If you work out the right hand side you'll see you get the original question back, maybe up to some numeric factors. Then you can try partial integration, which will move the d/dx to the x so maybe it becomes simpler. I don't know if it would help though, I think you'd still have to do $$\int e^{\sqrt{x}} \, \mathrm dx$$ in the end.

6. Jul 2, 2008

gamesguru

If you use my substitution, $u=\sqrt{x}, 2u du=dx$ then the integral becomes:
$$\int \sqrt{x}e^{\sqrt{x}}dx=\int u e^u 2u du=2\int u^2 e^u du$$.
Use integration by parts now, differentiate $u^2$ and integrate $e^u du$.
If you do the integration by parts correctly you get the final answer,
$$2e^{\sqrt{x}}(x-2\sqrt{x}+2)$$.
Hopefully you get the same thing.

7. Jul 2, 2008

phyzmatix

I'm starting to wonder if there is perhaps a typing error in the question. What makes me wonder even more is that the question only counts 5 marks, so it probably isn't supposed to be a huge calculation. Or is that a dangerous assumption to make and first year Calculus really is supposed to be this tricky?

8. Jul 2, 2008

phyzmatix

Oh, ok, I see now. I'm going to give that a go. Ridiculous as this sounds, I wasn't sure if I could combine the two approaches and typically, on hindsight, it suddenly seems the logical thing to do

9. Jul 3, 2008

phyzmatix

gamesguru, using your proposed substitution, I get

$$u = \sqrt{x}, 2\sqrt{x}du = dx$$

as you obviously did, however, I'm struggling to convince myself that it's mathematically correct to substitute into the substitution, i.e. to write

$$2\sqrt{x}du = dx$$

as

$$2udu = dx$$

I'm sorry if I'm being irritating, but I really want to understand this and how it works.

10. Jul 3, 2008

HallsofIvy

Of course that's correct- it's just algebra.

11. Jul 3, 2008

phyzmatix

Thank you for your input. I just wanted confirmation since I didn't find any examples in my textbook or study guide that did the same, which caused some doubt (I'm studying through a distance learning university, so I'm very reliant on examples as a means of explanation and to determine what's allowed and what isn't).

Thanks again!

Last edited: Jul 3, 2008
12. Jul 3, 2008

gamesguru

Here's the full reason for that.
We make the substitution $u=\sqrt{x}$.
Differentiate both sides with respect to x,
$$\frac{du}{dx}=\frac{1}{2\sqrt{x}}$$.
With me so far? Bring the dx over,
$$du=\frac{1}{2\sqrt{x}}dx$$.
Now we recognize the original substitution $u=\sqrt{x}$:
$$du=\frac{1}{2u}dx$$.
Finally, bring over the 2u:
$$2u du=dx$$.
That's the reason why, and you can check this by integration, 2u du->u^2 and dx->x so:
$$u^2=x, u=\sqrt{x}$$.
Hope that helps.