Some impossible integrals...

Hy everyone!

I don't know exactly how to use some symbols here, but i will denote with int(function) as being the integral of a function.

So I have 4 problems, each of them will be very helpful if you can give a suggestion somehow...

1) lim (n tends to infinity) of n^2 * ∫∫((x^2-y^2)/4)^n dx dy.

The first integral is defined from 0 to 1 (and the variables is y, and the second from y to 2-y and the variable is x).

There is also a hint given: change variables by rotating the axes through an angle of pi/4.

2) ∫∫e^(2x-x^2) dx dy. The first integral is defined from 0 to 1 and the variable is y, the second is defined from 0 to 1-y(1/3) and the variable is x.

3) ∫ 1/((1+x^2)*(1+x^a)) dx, where a is a constant. The integral is defined from 0 to infinity, and the variable is x.

4) ∫f * (x-1/x) dx, where f is a probability density function, the integral is defined from minus infinity to plus infinity and the variable is x.

Thank you a lot if you manage to help me somehow.

Re: Some impossible integrals...

Quote:

Originally Posted by

**TibiSam** Hy everyone!

I don't know exactly how to use some symbols here, but i will denote with int(function) as being the integral of a function.

So I have 4 problems, each of them will be very helpful if you can give a suggestion somehow...

1) lim (n tends to infinity) of n^2 * ∫∫((x^2-y^2)/4)^n dx dy.

The first integral is defined from 0 to 1 (and the variables is y, and the second from y to 2-y and the variable is x).

There is also a hint given: change variables by rotating the axes through an angle of pi/4.

2) ∫∫e^(2x-x^2) dx dy. The first integral is defined from 0 to 1 and the variable is y, the second is defined from 0 to 1-y(1/3) and the variable is x.

3) ∫ 1/((1+x^2)*(1+x^a)) dx, where a is a constant. The integral is defined from 0 to infinity, and the variable is x.

4) ∫f * (x-1/x) dx, where f is a probability density function, the integral is defined from minus infinity to plus infinity and the variable is x.

Thank you a lot if you manage to help me somehow.

For number 2, I would suggest reversing the order of integration, as can not be evaluated.

Your integration limits give and . Some rearranging gives and , so your integral is

Now make the substitution , note that when and when and also that

then the integral becomes

Re: Some impossible integrals...