Solve Double Integration: \int\frac{x^2-y^2}{(x^2+y^2)^2}dx

[sadris]

$$\int \frac{x^2 - y^2}{(x^2 + y^2)^2} dx$$

Above is an integration involved in a double integration, I know the answers via TI-89, but I am trying to find out how to get them I have tried trig substitution, u sub, integration by parts, etc. etc. And I am out of ideas. Can anyone please help?

Thanks!

 

[StatusX]

Do you know partial fractions?

 

[jpr0]

First of all, label your original integral as, say, $I$

$$I=\int \frac{x^2 - y^2}{(x^2 + y^2)^2} dx$$

Then split the integral into two separate integrals, one where $x^2$ is in the numerator, and other where $y^2$ is in the numerator.

With the first integral (the numerator $x^2$ one), integrate this by parts with the aim of maxing the denominator of your integral $(x^2 + y^2)$.

After you have done this, look at the complete expression for $I$ (substituting into this the integration-by-parts you just carried out).The integrals which remain (the one involving $y^2$ in the numerator, and the integral which is left from integration by parts), combine them both back into one single integral, with the integrand expressed as a single fraction. Compare this integral with the the expression for $I$ on the first line (i.e. the expression above). You should then have an algebraic equation in $I$, which you can solve.

 

[wurth_skidder_23]

Do a trig substitution with x = y Tan[theta], dx = y (Sec[theta])^2. This will reduce your function to something resembling a trig identity that can easily be integrated.

 

[jpr0]

Partial fractioning is probably the easiest way as StatusX suggested.

 

[sadris]

Do you know partial fractions?

Yea I tried that but you end up with

$$x^2 - y^2 = A(x^2 + Y^2) + B(x^2 + Y^2)$$

And when you set x^2 = -y^2 you would end up with -2y^2 =0 which really isn't a helpful expression :(

 

[StatusX]

Why would you set x^2=-y^2? You need to write:

$$\frac{x^2-y^2}{(x^2+y^2)^2} = \frac{Ax+B}{x^2+y^2}+\frac{Cx+D}{(x^2+y^2)^2}$$

Then solve for A,B,C,D.

 

[jpr0]

There's a quicker way to manipulate the integrand into a form which ia easier to integrate if you say that

$$(x^2 + y^2) = (x+iy)(x-iy)$$

and

$$x^2 - y^2 = \frac{1}{2}\left[(x+iy)^2 + (x-iy)^2\right]$$

So,

$$\frac{x^2 - y^2}{(x^2 + y^2)^2} = \frac{1}{2}\frac{(x+iy)^2 + (x-iy)^2}{(x+iy)^2(x-iy)^2}$$