Could you help me with this improper integral

Question

How can I evaluate this improper integral?

$$\displaystyle\int_0^{\infty}\frac{1}{x(1+x^2)}\,dx $$

Answer 1

An indefinite integral leads to (breaking it into simple fractions) $$log(x)-\frac{1}{2}log(1+x^2)$$ So clearly you have problems at $x=0$.

Answer 2

Help getting started: Start by writing the integral like this: $$ \int_0^\infty \frac{1}{x(1+x^2)}\; dx = \lim_{t\to 0^+}\int_t^1\frac{1}{x(1+x^2)}\; dx + \lim_{s\to \infty}\int_1^{s}\frac{1}{x(1+x^2)}\; dx. $$ If you can show that one of these integrals is divergent, then the original integral is divergent.

First we can consider the integral $$\begin{align} \int_t^1 \frac{1}{x(1+x^2)}\; dx &= \int_t^1 \frac{1}{x} - \frac{x}{1+x^2}\; dx \\ &= \ln\lvert x\rvert - \frac{1}{2}\ln\lvert 1 + x^2\rvert\left. \right]_t^1\\ &= \ln\left(\frac{x}{\sqrt{1+x^2}}\right)\left.\right]_t^1 \end{align} $$ I believe that when you evaluate this and take the limit as $t$ goes to $0$ from the right, then is $-\infty$. So yes, the integral is divergent.

But, you should probably check the details.

Answer 3

$$\int_0^1\frac{1}{x(1+x^2)}\,dx\geq\int_0^1\frac1{2x}\,dx$$ which is more clearly divergent ($\lim\limits_{t\to0^+}\left[\frac{1}{2}\ln(x)\right]_t^1$).

Answer 4

Try this:

$$x=\tan { u } $$ $$dx=\frac { 1 }{ { \left( \cos { u } \right) }^{ 2 } } du$$ $$\int _{ 0 }^{ \infty }{ \frac { 1 }{ x\left( { x }^{ 2 }+1 \right) } dx } =\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \cos { u } }{ \sin { u } } du } $$