1. question about improper integral

i could not find that integral converges or disconverges, please help me!

the lower bounds of integral is 1 and the upper bound is infinite and f(x)=arctan(1/x^2). i am sorry that i don't know if there is any code to write it in mathematical expression

2. Originally Posted by gilgames
i could not find that integral converges or disconverges, please help me!

the lower bounds of integral is 1 and the upper bound is infinite and f(x)=arctan(1/x^2). i am sorry that i don't know if there is any code to write it in mathematical expression
$\displaystyle \int_{1}^{\infty}\tan^{-1}\left( \frac{1}{x^2}\right)dx$

Integration by parts with $\displaystyle u=\tan^{-1}\left( \frac{1}{x^2}\right)$ and $\displaystyle dv=dx$ gives

$\displaystyle x\tan^{-1}\left( \frac{1}{x^2}\right)+\int_{1}^{\infty} \frac{2x^2}{x^4+1}dx < x\tan^{-1}\left( \frac{1}{x^2}\right)+\int_{1}^{\infty} \frac{2}{x^2}dx$

Since the right hand side converges by comparison the left hand side must also

3. thanks for the response. i had tried the partial integration but could not find the result. thanks