Improper integral Question 18. For [ Dn=left(x, y) in...

Improper integral Question 18. For [ D_{n}=left{(x, y) in mathbb{R}^{2}: x^{2}+y^{2} leq n^{2} ight}, ] compute the following integral [ lim _{n ightarrow infty} iint_{D_n} frac{log left(1+x^{2}+y^{2} ight)}{left(1+x^{2}+y^{2} ight)^{2}} d x d y ]

Transcript

00:01 So here we have dn equals the set of xy such that s squared plus y squared is less than or equal to n squared.

00:07 And we want to compute the integral of the limit as n approaches infinity.

00:11 The double integral of d of log of 1 plus x squared plus y squared over 1 plus x squared plus y squared squared of dx dy.

00:23 So let's go ahead and rewrite using polar coordinates.

00:27 So this is going to equal, for the dn part, the double integral, where r goes from 0 to n, and theta goes from 0 to 2 pi, of r times log of 1 plus r squared over 1 plus r squared to the power of 2.

00:51 So now we can go ahead and integrate this.

00:56 So we're going to let u equal 1 plus r squared so du equals 2r dr so dr is going to equal 1 over 2r du so we then have the integral of log of u over u squared times 1 over 2 du u and our new bounds are going to go from 1 to 1 plus n squared so then here we're gonna go ahead and use integration by parts so what we get is we're gonna let u or sorry v equal log of u and dw equal u to the negative 2 du so then dv dv equals 1 over u, and dw is then going to equal negative u to the negative 1.

01:54 So what this integral ends up becoming is negative log u over u minus 1 over u.

02:06 And now we need to go ahead and apply our limits of integration with the factor of 1 half out in the front.

02:12 So from 1 to 1 plus n squared, and this is going to equal 1 half times negative log of 1 plus n squared over 1 plus n squared minus 1 over n squared plus 1 plus log of 1 plus 1...

Question

Improper integral Question 18. For [ D_{n}=left{(x, y) in mathbb{R}^{2}: x^{2}+y^{2} leq n^{2} ight}, ] compute the following integral [ lim _{n ightarrow infty} iint_{D_n} frac{log left(1+x^{2}+y^{2} ight)}{left(1+x^{2}+y^{2} ight)^{2}} d x d y ]

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