Using Parseval's Theorem to evaluate an integral -- Help please

[KeithKp]

Homework Statement

By applying Parseval's (Plancherel's) theorem to the function

are given by:

f(x) = -1 for -2 ≤ x < 0
1 for 0 ≥ x < 2
0 otherwise

determine the value of the following integral.

∫ dk sin^4(k)/(k^2) (Integral between ±infinity)

Homework Equations

Parseval's Theorem, i.e the integral of the modulus squared of a function is equal to the integral of the modulus squared of its Fourier transform.

Fourier Transform formula.

The Attempt at a Solution

I've tried multiple times to try and arrive at the correct answer of pi/2 but I just can't do it. Is the Fourier transform:

∫-dx e^(ikx) + ∫ dx e^(ikx)? (First integral is between -2 and 0, second between 0 and 2). Because I can't get the correct answer doing that.

And for the equation of the function, is it just sgn(x), with the integral between -2 and 2? How else do I write f(x) given those boundaries? If I use sgn(x) between -2 and 2, I get 4 for the left hand side. But I still can't derive that integral.

If someone would just start me off with the correct Fourier transform equation, i'll go away and crunch the integration myself.

Thank you.

 

[Svein]

(First integral is between -2 and 0, second between 0 and 2)

Yes, but then you need to adjust the exponential factor, since e^ikx is periodic with period 2π, not 4. Try e^ikπx/2.