Integrating $\frac{x^2}{(1+x^2)^3}$ Over the Real Line

[Fermat1]

integrate $\frac{x^2}{(1+x^2)^3}$ over the real line

 

[chisigma]

integrate $\frac{x^2}{(1+x^2)^3}$ over the real line

[sp]The function has in the upper half plane a pole of order 3 in z=i, so that is...

$\displaystyle \int_{- \infty}^{+ \infty} \frac{x^{2}}{(1 + x^{2})^{3}}\ d x = 2\ \pi i\ \lim_{z \rightarrow i} \frac{1}{2}\ \frac {d^{2}}{d z^{2}}\ \frac{z^{2}}{(z+i)^{3}} = \frac{\pi}{8}$[/sp]


Kind regards

$\chi$ $\sigma$

 

[Fermat1]

[sp]The function has in the upper half plane a pole of order 3 in z=i, so that is...

$\displaystyle \int_{- \infty}^{+ \infty} \frac{x^{2}}{(1 + x^{2})^{3}}\ d x = 2\ \pi i\ \lim_{z \rightarrow i} \frac{1}{2}\ \frac {d^{2}}{d z^{2}}\ \frac{z^{2}}{(z+i)^{3}} = \frac{\pi}{8}$[/sp]


Kind regards

$\chi$ $\sigma$

that is a way I had no really expected (although it is correct). What about a tan substitution

 

[Saitama]

integrate $\frac{x^2}{(1+x^2)^3}$ over the real line

Is this a homework problem?

Spoiler

Suspecting that this is a homework problem, I leave a hint, use $x=\tan\theta$. The resulting integral is straightforward.

 

[CellCoree]

I would approach this problem by first understanding the function we are integrating. The function $\frac{x^2}{(1+x^2)^3}$ is a rational function, meaning it is a ratio of two polynomials. It also has a singularity at $x=0$ since the denominator becomes zero at that point.

Next, I would consider the limits of integration, which in this case is over the real line. This means we need to integrate from $-\infty$ to $\infty$. This is known as an improper integral, since the limits extend to infinity.

To solve this integral, I would use techniques such as substitution or partial fractions to simplify the function and make it easier to integrate. I would then evaluate the integral using the Fundamental Theorem of Calculus or other integration rules.

However, since the function has a singularity at $x=0$, we need to be careful when evaluating the integral. We would need to use a technique called Cauchy's principal value, which involves taking the limit as the singularity approaches $x=0$ from both sides.

Overall, integrating $\frac{x^2}{(1+x^2)^3}$ over the real line is a challenging problem, but with proper techniques and careful consideration of the singularity, it can be solved. This integral may have applications in physics, engineering, or other scientific fields, so understanding how to solve it is important for further research and analysis.