Value of integrals with (x² + b²) raised to powers 2 and 4

[PowerWill]

Left my integral table at home, could someone tell me the value of these integrals?
$$\int_0^{\infty} \frac{dx}{(x^2+b^2)^2}$$

$$\int_0^{\infty} \frac{x^2}{(x^2+b^2)^2} dx$$
and the same as the latter but with the denominator raised to the power 4. Thanks!

 

[johnw188]

Left my integral table at home, could someone tell me the value of these integrals?
$$\int_0^{\infty} \frac{dx}{(x^2+b^2)^2}$$
$$\int_0^{\infty} \frac{x^2}{(x^2+b^2)^2} dx$$
and the same as the latter but with the denominator raised to the power 4. Thanks!

Just solve them yourself, they're both pretty standard trig substitutions. You should always solve the integral forms from the table yourself, at least once, before you use them. They're a nice tool for saving yourself time and effort, not a crutch to avoid learning your integration techniques properly.

 

[benorin]

www.integrals.com

 

[PowerWill]

What do I do with the arctan at infinity? Do I just use $$\frac{\pi}{2}$$ or do I have to be saucy about it?

 

[HallsofIvy]

Well, I wouldn't recommend be "saucy" about homework! Technically, you should take the limit of arctan t as t goes to $\infty$ but that is, of course, $\frac{\pi}{2}$.

 

[benorin]

EDIT: I was going to post the below, but I just realized the it doesn't matter what $\mbox{sgn}(b)$ is, assuming $b\neq 0$, take it to be positive since b is only given by b², we may assume it is positive.

For $$\int_0^{\infty} \frac{dx}{(x^2+b^2)^2}=\lim_{M\rightarrow\infty} \frac{1}{2b^3} \left(\frac{bM}{b^2+M^2} + \tan^{-1}\left( \frac{M}{b}\right) -0\right)$$
$$=0+\lim_{M\rightarrow\infty} \frac{1}{2b^3} \tan^{-1}\left( \frac{M}{b}\right)=\mbox{sgn}(b) \frac{\pi}{4b^3}$$