# Evaluating an Integral

• Feb 25th 2011, 12:17 PM
kikiya
Evaluating an Integral

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, 12:20 PM
TheEmptySet
Quote:

Originally Posted by kikiya

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 five me some insight on this! (Wondering)

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

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

If you sub this in you should get

$\displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int e^{u}du$
• Feb 25th 2011, 12:21 PM
pickslides
You're substitution is correct, what did you find $\displaystyle \frac{du}{dx}$ to be?

Edit: TES beat me to the punch.
• Feb 25th 2011, 12:24 PM
pickslides
Quote:

Originally Posted by TheEmptySet

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

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

If you sub this in you should get

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

I get

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

$\displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int e^{u}du$
• Feb 25th 2011, 12:30 PM
TheEmptySet
Quote:

Originally Posted by pickslides
I get

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

$\displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int e^{u}du$

and you are correct!
• Feb 25th 2011, 12:38 PM
kikiya
I'm sorry, but where did you get the 2 that's outside of the integral? Please explain your process lol
• Feb 25th 2011, 12:58 PM
pickslides
O.K given $\displaystyle u = \sqrt{x}$what do you get for $\displaystyle \frac{du}{dx}$ ?
• Feb 25th 2011, 02:48 PM
kikiya
Sorry I'm late but is it 1/ 2 (x^1/2) ?
• Feb 25th 2011, 03:25 PM
NOX Andrew
Exactly. Unfortunately, there isn't a 1/2 inside the integral, so you can't substitute $du = \frac{1}{2\sqrt{x}}dx$ 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:

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

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

$2\int e^u \, du$
• Feb 26th 2011, 04:06 AM
HallsofIvy
Quote:

Originally Posted by kikiya

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

Strictly speaking, what you wrote here was
$\frac{\left(e^x)^{1/2}}{x^{1/2}}= \frac{e^{\frac{x}{2}}}{x^{1/2}}$
but apparently you meant
$e^{x^{1/2}}}{x^{1/2}}$
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, 09:05 AM
kikiya
Oh, I see the error! Thank you! :)
• Feb 26th 2011, 09:07 AM
kikiya
Thank you for walking me through it! I was confiused about the balancing out, but now I get it! Thank you! :D