I need to find k such that the integral from -infinity to infinity equals 1. I know the answer is 4 for the eqn (see picture), but my work doesn't agree. Help?

Question

anonymous · Answer

This is the problem

turingtest · Answer

unfortunately latex is down so this will be a bit of a pain to type :/

anonymous · Answer

That's okay. So far I've been able to separate the integral so that I have 1 = 2k lim(b--> infinity) of integral (0 to b) of |x|/(x^2 +4) dx

anonymous · Answer

Hi TuringTest!

anonymous · Answer

I know that I need do a u-sub. So I've gotten so far as the integral (blah) of 1/u^2 du

anonymous · Answer

times k equals 1. But when I solve out, it doesn't work. I don't get four.

turingtest · Answer

yeah I am having trouble too I think...

anonymous · Answer

also, can I really do that? should the k be multiplied by 2, or should the integral be rewritten without it, just the integral from 0 to b. (lim as b approaches infinity)

turingtest · Answer

you have to split the integral because of the absolute value

turingtest · Answer

f(x)=-kx/(x^2+4)^2  for x<0
f(x)=kx/(x^2+4)^2  for  x>=0

anonymous · Answer

right. So in the integral, I made up for that by multiply by two, reasoning that the distance from 0 to infinity is the same as the distance from -infinity to 0

anonymous · Answer

eventually, with u-sub (if I'm doing it right) the 2 on the outside of the integral will cancel out.

turingtest · Answer

give me a minute to work it on paper...

anonymous · Answer

okay, I'll do the same

turingtest · Answer

okay I got it :)

turingtest · Answer

gonna be a pain to type though

anonymous · Answer

you did! What was I doing wrong?

turingtest · Answer

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