1. ## Integral

Evaluate
$
\int_{0}^1\frac{2x-(1+x^2)^2\cot^{-1}x}{(1+x^2)\{(1-(1+x^2)\cot^{-1}x\}}dx
$

2. Originally Posted by pankaj
Evaluate
$
\int_{0}^1 \frac{2x-(1+x^2)^2\cot^{-1}x}{(1+x^2)\{1-(1+x^2)\cot^{-1}x\}}dx
$
man, this is one ugly integral! haha ...

ok, we have this identity: $\frac{2x-(1+x^2)^2\cot^{-1}x}{(1+x^2)\{1-(1+x^2)\cot^{-1}x\}}=1 + \frac{2x}{1+x^2} - \frac{1-2x \cot^{-1} x}{1-(1+x^2)\cot^{-1}x}.$ we also have $\frac{d}{dx}(1 - (1+x^2) \cot^{-1}x) = 1 - 2x \cot^{-1} x.$

thus: $\int \frac{2x-(1+x^2)^2\cot^{-1}x}{(1+x^2)\{1-(1+x^2)\cot^{-1}x\}} \ dx=x + \ln \left|\frac{1+x^2}{1 - (1+x^2)\cot^{-1} x} \right| + C.$ the value of your definite integral is $1+\ln 2.$

3. Originally Posted by NonCommAlg
man, this is one ugly integral! haha ...

I am hungry for your beautiful integrals ....

4. Here is my solution:

$
\int_{0}^1\frac{\frac{2x}{(1+x^2)^2}-\cot^{-1}x}{\frac{1}{1+x^2}-\cot^{-1}x}dx$

= $
\int_{0}^1\frac{\frac{2x}{(1+x^2)^2}-\frac{1}{1+x^2}+\frac{1}{1+x^2}-\cot^{-1}x}{\frac{1}{1+x^2}-\cot^{-1}x}dx
$

= $
-\int_{0}^1\frac{\frac{-2x}{(1+x^2)^2}+\frac{1}{1+x^2}}{\frac{1}{1+x^2}-\cot^{-1}x}dx+\int_{0}^11dx
$

= $\ln 2 +1$

5. Originally Posted by pankaj
Here is my solution:

$
\int_{0}^1\frac{\frac{2x}{(1+x^2)^2}-\cot^{-1}x}{\frac{1}{1+x^2}-\cot^{-1}x}dx$

= $
\int_{0}^1\frac{\frac{2x}{(1+x^2)^2}-\frac{1}{1+x^2}+\frac{1}{1+x^2}-\cot^{-1}x}{\frac{1}{1+x^2}-\cot^{-1}x}dx
$

= $
-\int_{0}^1\frac{\frac{-2x}{(1+x^2)^2}+\frac{1}{1+x^2}}{\frac{1}{1+x^2}-\cot^{-1}x}dx+\int_{0}^11dx
$

= $\ln 2 +1$
this is better than my solution!