How do you evaluate the inverse transform below using convolution ?
$$
\mathcal{ L^{-1} } \left[ {\frac{s}{(s^2 + a^2)^2}} \right]
$$
I tried
$$\begin{align} \mathcal{ L^{-1} } \left[ {\frac{s}{(s^2 + a^2)^2}} \right](t) &=
\int\limits_0^t \sin t \cdot \cos(at – a\tau) \, d\tau = \sin t \cdot \int\limits_0^t \cos(at – a\tau) \, d\tau \\
&= \sin t \cdot \left[ -\frac{1}{a} \sin(at – a\tau) \right]_0^t = \sin t \cdot \left[ 0 – \left(-\frac{1}{a} \sin(at) \right) \right] \\ &= \frac{1}{a} \sin^2(at) \end{align}
$$
What have I done wrong? The answer in my book is
$$
\frac{t}{2a} \sin(at)
$$
Best Answer
Assume someone tells you to derivative $ \frac{1}{(s^2+a^2)}$ respect to $s$ wherein $a$ is a constant. You certainly will reply $$\frac{-2s}{(s^2+a^2)^2}$$ $$\left(\frac{1}{(s^2+a^2)}\right)'=\frac{-2s}{(s^2+a^2)^2}$$ or $$\left(\frac{-1}{2(s^2+a^2)}\right)'=\frac{s}{(s^2+a^2)^2}$$ or $$\frac{1}{2}\times(-1)^1\left(\frac{1}{(s^2+a^2)}\right)'=\frac{s}{(s^2+a^2)^2}$$ But surely know that $$\mathcal{L}(t^1f(t))=(-1)^1F'(s)$$. Now can you find the proper $f(t)$ form tow last equalities when you know that $$\mathcal{L}\left(\frac{1}{a}\sin(at)\right)=\frac{1}{s^2+a^2}$$. This another approach besides to yours and @saz's.