ChaoticLlama
Junior Member
- Joined
- Dec 11, 2004
- Messages
- 199
I've been working out of a textbook teaching myself integration, and I've come upon something that the book does a poor job of explaining (re: doesn't).
My question is, how do I integrate this?
\(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )^2 }}}\)
I of course know \(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )}}} = \arctan (x) + c\)
My confusion comes from, they try to make a point, yet I fail to get it. They say for one of their examples:
We want to find the integral:
\(\displaystyle I_n = \int {\frac{{dx}}{{(1 + x^2 )^n }}}\)
They follow up by saying, let us deal with the case of n=1.
And instead of just saying the integral is arctan(x), they integrate by parts and establish in a round-about way that
\(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )^2 }}} = \frac{1}{2}\left( {\frac{x}{{1 + x^2 }} + \arctan (x)} \right)\)
Is there any way to integrate this function directly? Your help is appreciated.
My question is, how do I integrate this?
\(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )^2 }}}\)
I of course know \(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )}}} = \arctan (x) + c\)
My confusion comes from, they try to make a point, yet I fail to get it. They say for one of their examples:
We want to find the integral:
\(\displaystyle I_n = \int {\frac{{dx}}{{(1 + x^2 )^n }}}\)
They follow up by saying, let us deal with the case of n=1.
And instead of just saying the integral is arctan(x), they integrate by parts and establish in a round-about way that
\(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )^2 }}} = \frac{1}{2}\left( {\frac{x}{{1 + x^2 }} + \arctan (x)} \right)\)
Is there any way to integrate this function directly? Your help is appreciated.