Question

Evaluate the integral $\int \frac{1}{1+x^2}dx$ from 0 to $\infty$

Answer 1

We know that $\int \frac{1}{1+x^2}dx=arc\tan x+C.$

Then, by the Fundamental Principle of Definite Integral,

$\therefore I={\int }_0^{\infty }\frac{1}{1+x^2}dx=\underset{y\to \infty }{lim}{\int }_0^y\frac{1}{1+x^2}dx$

$=\underset{y\to \infty }{lim}{\left[arc\tan x\right]}_0^y$

$=\underset{y\to \infty }{lim}[arc\tan y-arc\tan 0]$

Since, the $arc\tan$ fun. is continuous on $R$, we have,

$I=arc\tan \left\{\underset{y\to \infty }{lim}y\right\}-0$

$=\frac{\pi }{2}$.

π/2 is the solution for given integral equation.