Delta function/fourier transform

CalcYouLater

1. The problem statement, all variables and given/known data
Hello. This question is about Fourier transforms and the Delta function.

Find the fourier transform of:

[tex]g(k)=\frac{10sin(3k)}{k+\pi}[/tex]



2. Relevant equations

[tex]f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}g(k)e^{(ikx)}dk[/tex]

[tex]\delta(x-\acute{x})\doteq\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ik(x-\acute{x})}dk[/tex]

3. The attempt at a solution

I began to solve this problem and quickly got to a point where I saw the above mentioned delta function representation. Using Euler's formula, I re-wrote the sine term. My question is, can I use the delta function here:

[tex]f(x)=\frac{5}{i}\int_{-\infty}^{\infty}\frac{e^{ik(3+x)}}{k+\pi}-\frac{e^{ik(x-3)}}{k+\pi}dk[/tex]

Now that I have "said that outloud" it seems like the answer is of course not. I guess really I am just looking for a short cut to get out of doing a lot of math. Is there an efficient way of evaluating this integral without "trudging" through it?