Appropriate Change of Variables for integration

[Flyboy27]

Can anyone give me any hints as to find a suitable change of variables for this integral.

infinity
/
|dt/(a^2+t^2)^3/2 =
|
/ -infinity


=2/a^2 * integral below
Pi/2
/
| cos t dt
|
/ 0


Thank you in advance

 

[Jameson]

$$\int_{-\infty}^{\infty}\frac{dt}{(a^2+t^2)^\frac{3}{2}} = \frac{2}{a^2} * \int_{0}^{\frac{\pi}{2}} \cos{t}dt$$

Is this correct?

I think you can do a $\tan^{-1}$ substitution and use triangles to rewrite the integral.

 

[Hurkyl]

Shouldn't it be the same change of variable as for

$$\int \frac{dx}{a^2 + x^2}$$

or

$$\int \sqrt{a^2 + x^2} \, dx$$

?

 

[dextercioby]

I've always supported hyperbolic trig.functions used in substitutions.In your case,it's ~$$\sinh x$$...

Daniel.

 

[Flyboy27]

Alright using a table of integrals and some algebra here is what I have so far:

$$\int_{-\infty}^{\infty}\frac{dt}{(a^2+t^2)^\frac{3}{2}} = \int_{-\infty}^{\infty}\frac{a}{t^2(a^2+t^2)^\frac{3}{2}} + \frac{3}{t^2}}\int_{-\infty}^{\infty}\frac{dt}{(a^2+t^2)^\frac{1}{2}}$$


Am I getting anywhere...I don't think so...

 

[dextercioby]

How did u pull that square 't' outta the integral...?

Daniel.

 

[Flyboy27]

I used a table of integrals...and some simple algebra, unless I looked at the wrong intergral form, but I don't think I did, so anyway, where do I use the substitution?