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I am having some trouble with knowing the best route to go about evaluating this integral for the I.F.T. It states the following.

$w(t)$ has the Fourier Transform $W(f)= \dfrac{(j\pi f)}{(1+j2\pi f)}$

Given $w_1(t) = w(t/5)$ we need to find the spectrum of $w(t)$.

So to start, we need to know what $w(t)$ is, and this can be done by taking the I.F.T. to find out what $w$ is and then we would be able to plug in $t=t/5$ to find the F.T. of that to get the spectrum for $w_1(t)$. The trouble I am having is evaluating the integral for the I.F.T. This is what I have done so far.

$$ \begin{align} \displaystyle w(t) &= \int_{-\infty}^{\infty} \! W(f)e^{j\omega t}\,\mathrm{d}f \\ &= \int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}e^{j2\pi ft}\,\mathrm{d}f, \text{ where } \omega=2\pi f \\ &= \int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}\Big[\cos(2\pi f t)+j\sin(2\pi ft) \Big]\,\mathrm{d}f \end{align} $$

From here, I do not know the best way to go about integrating the complex component with the complex exponential or trig functions.

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You can find the answer directly by using the scaling property of the Fourier Transform, namely $W_1(f) = 5W(5f)$. – Tpofofn Mar 23 '12 at 3:22
@Tpofofn: Thanks, I just realized I did not need to find the inverse to get $w_1(t)$ because it was only asking for the frequency spectrum. But, how would I find the inverse of $W(f)$ to get what $w(t)$ is. Either using the definition which I started, or transform properties. I managed to get this from using the properties but not sure if it is correct: $w(t)= 1/2e^{-t}\frac{\mathrm{d}w}{\mathrm{d}t} u(t)\,$ where $u(t)$ is the unit step function. – night owl Mar 23 '12 at 4:44
"..not sure if it is correct..." Hint: Try taking the forward Fourier transform of what you think $w(t)$ is to see if you get $W(f)$. – Dilip Sarwate Mar 23 '12 at 12:33
$\frac{d}{dt}(w(t)e^t)=\frac{d}{dt}(\int_{-\infty}^{\infty} \! \frac{j\pi f}{1+j2\pi f}e^{(j2\pi f+1)t} df)$ you can try that way to cancel $1+j2\pi f$ and to do simple form of integral? – Mathlover Mar 23 '12 at 15:35
@Mathlover: Thanks. So when doing this, I get for the right hand side before evaluating the limits is: $\frac{e^{t+2 i \pi f t} (2 \pi f t+i)}{4 \pi t^2}$. Now, plugging in limits, the integral will not converge on $[-\infty,\infty]\,$. How do we go from here to fully obtain the inverse of $W(f)$? – night owl Mar 23 '12 at 20:08

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