# Not sure how to do this integral.

• Apr 16th 2008, 04:04 PM
Gotovina7
Not sure how to do this integral.
Here she is.

$\displaystyle \int \frac {1}{x^3}e^\frac{1}{x}$

Thanks.
• Apr 16th 2008, 04:30 PM
Mathstud28
Quote:

Originally Posted by Gotovina7
Here it is.

$\displaystyle \int \frac {1}{x^3}e^\frac{1}{x}$

Thanks.

Using power series rewrite it as $\displaystyle \int\frac{\sum_{n=0}^{\infty}\frac{\bigg(\frac{1}{ x}\bigg)^n}{n!}}{x^3}dx\Rightarrow\int\frac{\sum_{ n=0}^{\infty}\frac{x^{-n}}{n!}}{x^3}dx$

then rewrite it as $\displaystyle \int\sum_{n=0}^{\infty}\frac{x^{-n-3}}{n!}dx$

integrating we get $\displaystyle \int{\frac{e^{\frac{1}{x}}}{x^3}}dx=\sum_{n=0}^{\i nfty}\frac{x^{-n-2}}{(-n-2)n!}$
or you could just use parts whichever way your teacher wants
• Apr 16th 2008, 04:31 PM
mr fantastic
Quote:

Originally Posted by Gotovina7
Here it is.

$\displaystyle \int \frac {1}{x^3}e^\frac{1}{x}$

Thanks.

Make the obvious substitution $\displaystyle u = \frac{1}{x} \Rightarrow dx = -x^2 \, du$.

Then the integral becomes $\displaystyle \int u e^u \, du$.

Now use integration by parts.
• Apr 16th 2008, 04:59 PM
Gotovina7
Ooh, I got it now! Thanks guys!
• Apr 16th 2008, 05:44 PM
Gotovina7
How would I do this one?

$\displaystyle \int \frac{1}{e^x+2 \sqrt{e^x}+1}dx$

Thanks :D
• Apr 16th 2008, 05:50 PM
Mathstud28
Quote:

Originally Posted by Gotovina7
How would I do this one?

$\displaystyle \int \frac{1}{e^x+2 \sqrt{e^x}+1}dx$

Thanks :D

Next time start a new post but does this help $\displaystyle \int\frac{1}{e^{x}+2e^{\frac{x}{2}}+1}dx=\int\frac {1}{(e^{\frac{x}{2}}+1)^2}dx$?
• Apr 16th 2008, 05:52 PM
mr fantastic
Quote:

Originally Posted by Gotovina7
How would I do this one?

$\displaystyle \int \frac{1}{e^x+2 \sqrt{e^x}+1}dx$

Thanks :D

Note: New question => New thread.

Make the obvious substitution $\displaystyle u^2 = e^x \Rightarrow 2u \frac{du}{dx} = e^x = u^2 \Rightarrow dx = \frac{2 \, du}{u}$.

Then the integral becomes (after a little bit of algebra): $\displaystyle 2 \int \frac{du}{(u+1)^2 u}$.

Now use partial fractions etc.
• Apr 16th 2008, 05:55 PM
Gotovina7
Ooh, I see it now. Thanks guys. Sorry about the post.
• Apr 16th 2008, 06:14 PM
Jhevon
Quote:

Originally Posted by Gotovina7
Need help with another one, scroll down :D