Integration

ChaoticLlama

Junior Member
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Dec 11, 2004
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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.
 
ChaoticLlama said:
My question is, how do I integrate this?
\(\displaystyle \int {\frac{{dx}}{{(1 + x^2 )^2 }}}\)
Let x = tan(u).
 
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