Please help me evaluate

e^(x)^1/2/ (x^1/2)

I started out with u substitution. First I let radical x be u, but I couldn't work it out properly because I didn't know what to cross out. Please give me some insight on this! (Wondering)

Printable View

- Feb 25th 2011, 11:17 AMkikiyaEvaluating an Integral
Please help me evaluate

e^(x)^1/2/ (x^1/2)

I started out with u substitution. First I let radical x be u, but I couldn't work it out properly because I didn't know what to cross out. Please give me some insight on this! (Wondering) - Feb 25th 2011, 11:20 AMTheEmptySet
Your substitution is correct.

$\displaystyle \displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx$

$\displaystyle u=\sqrt{x} \implies du=\frac{dx}{2\sqrt{x}}$

If you sub this in you should get

$\displaystyle \displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int e^{u}du$ - Feb 25th 2011, 11:21 AMpickslides
You're substitution is correct, what did you find $\displaystyle \displaystyle \frac{du}{dx}$ to be?

Edit: TES beat me to the punch. - Feb 25th 2011, 11:24 AMpickslides
- Feb 25th 2011, 11:30 AMTheEmptySet
- Feb 25th 2011, 11:38 AMkikiya
I'm sorry, but where did you get the 2 that's outside of the integral? Please explain your process lol

- Feb 25th 2011, 11:58 AMpickslides
O.K given $\displaystyle \displaystyle u = \sqrt{x}$what do you get for $\displaystyle \displaystyle \frac{du}{dx}$ ?

- Feb 25th 2011, 01:48 PMkikiya
Sorry I'm late but is it 1/ 2 (x^1/2) ?

- Feb 25th 2011, 02:25 PMNOX Andrew
Exactly. Unfortunately, there isn't a 1/2 inside the integral, so you can't substitute $\displaystyle du = \frac{1}{2\sqrt{x}}dx$ yet. Fortunately, you can place a 1/2 inside the integral as long as you place a 2 outside the integral (to balance the 1/2). Now, the problem looks like:

$\displaystyle 2\int \frac{e^{\sqrt{x}}}{2\sqrt{x}} \, dx$

Substituting $\displaystyle u = \sqrt{x}$ and $\displaystyle du = \frac{1}{2\sqrt{x}}dx$ gives:

$\displaystyle 2\int e^u \, du$ - Feb 26th 2011, 03:06 AMHallsofIvy
Strictly speaking, what you wrote here was

$\displaystyle \frac{\left(e^x)^{1/2}}{x^{1/2}}= \frac{e^{\frac{x}{2}}}{x^{1/2}}$

but apparently you meant

$\displaystyle e^{x^{1/2}}}{x^{1/2}}$

which would be e^(x^(1/2))/x^(1/2)

Quote:

I started out with u substitution. First I let radical x be u, but I couldn't work it out properly because I didn't know what to cross out. Please give me some insight on this! (Wondering)

- Feb 26th 2011, 08:05 AMkikiya
Oh, I see the error! Thank you! :)

- Feb 26th 2011, 08:07 AMkikiya
Thank you for walking me through it! I was confiused about the balancing out, but now I get it! Thank you! :D