You have that
$$\frac{1}{{\left( {1 + {u^2}} \right)\left( {1 + {a^2}{u^2}} \right)}} = \left( {\frac{A}{{1 + {u^2}}} + \frac{B}{{1 + {a^2}{u^2}}}} \right)$$
Thus you want (after cross mult.)
$$1 = A + A{a^2}{u^2} + B + B{u^2}$$
This is
$$\eqalign{
& A + B = 1 \cr
& A{a^2} + B = 0 \cr} $$
Which gives
$$A = \frac{1}{{1 - {a^2}}}$$
and in turn
$$B = 1 - A = \frac{{{a^2}}}{{{a^2} - 1}}$$
which means
$$\frac{1}{{\left( {1 + {u^2}} \right)\left( {1 + {a^2}{u^2}} \right)}} = \frac{1}{{{a^2} - 1}}\left( {\frac{{{a^2}}}{{1 + {a^2}{u^2}}} - \frac{1}{{1 + {u^2}}}} \right)$$
Can you move on?
A few ways to integrate $e^{-ax}\sin(x)$:
1) Integration by parts:
$$\begin{align}\int e^{-ax}\sin(x)~dx&=-e^{-ax}\cos(x)-a\int e^{-ax}\cos(x)~dx\\&=-e^{-ax}\cos(x)-a\left(e^{-ax}\sin(x)+a^2\int e^{-ax}\sin(x)~dx\right)\end{align}$$
Let $I=\int e^{-ax}\sin(x)~dx$ to see that
$$I=-e^{-ax}\cos(x)-a\left(e^{-ax}\sin(x)+a^2I\right)$$
which is a linear equation to solve for $I$.
2) Euler's formula:
This is a more complex method (get the pun?) but pretty straight forward. One may either use
$$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}\qquad or\qquad\sin(x)=\Im(e^{ix})$$
Using the second one for simplicity, we see that
$$\begin{align}I&=\Im\int e^{-ax}e^{ix}~dx\\&=\Im\int e^{(i-a)x}~dx\\&=\Im\left(\frac1{i-a}e^{(i-a)x}\right)+c\\&=\Im(u+vi)+c\\&=v+c\end{align}$$
where $v$ is the imaginary part of $\frac1{i-a}e^{(i-a)x}$.
As per the original problem, this is how I would've tackled it, using the complex method:
$$\begin{align}\int_0^1\frac{e^{-ax}\sin(x)}x\ dx&=\int_0^1e^{-ax}\sin(x)\int_0^\infty e^{-xt}\ dt\ dx\\&=\int_0^\infty\int_0^1e^{-(a+t)x}\sin(x)\ dx\ dt\\&=\int_0^\infty\Im\int_0^1e^{[i-(a+t)]x}\ dx\ dt\\&=\int_0^\infty\Im\left(\frac1{i-(a+t)}e^{[i-(a+t)]x}\bigg|_{x=0}^1\right)\ dt\\&=\int_0^\infty\frac1{1+(a+t)^2}\left(1-\frac{\cos(t)+(a+t)\sin(t)}{e^{a+t}}\right)\ dt\end{align}$$
And I think this is far as you can go this way.
Best Answer
Let's consider the integral
\begin{align}I(\alpha)&=\int_0^{\Large\frac{\pi}{2}}\frac{\ln\,(1+\cos\alpha\,\sin\,\phi)}{\sin\,\phi}\;d\phi\quad\Rightarrow\quad\phi\mapsto \frac{\pi}{2}-\phi\\ &=\int_0^{\Large\frac{\pi}{2}}\frac{\ln\,(1+\cos\alpha\,\cos\,\phi)}{\cos\,\phi}\;d\phi, \qquad 0 < \alpha < \pi.\end{align}
Differentiating $I(\alpha)$ with respect to $\alpha$, we have
\begin{align} {I}'(\alpha) &= \int_0^{\Large\frac{\pi}{2}} \frac{\partial}{\partial\alpha} \left(\frac{\ln(1 + \cos\alpha \cos \phi)}{\cos \phi}\right)\,d\phi \\ &=-\int_0^{\Large\frac{\pi}{2}}\frac{\sin \alpha}{1+\cos \alpha \cos \phi}\,d\phi \\ &=-\int_0^{\Large\frac{\pi}{2}}\frac{\sin \alpha}{\left(\cos^2 \frac{\phi}{2}+\sin^2 \frac{\phi}{2}\right)+\cos \alpha\,\left(\cos^2\,\frac{\phi}{2}-\sin^2 \frac{\phi}{2}\right)}\,d\phi \\ &=-\frac{\sin\alpha}{1-\cos\alpha} \int_0^{\Large\frac{\pi}{2}} \frac{1}{\cos^2\frac{\phi}{2}}\frac{1}{\left[\left(\frac{1+\cos \alpha}{1-\cos \alpha}\right) +\tan^2 \frac{\phi}{2} \right]}\,d\phi \\ &=-\frac{2\,\sin\alpha}{1-\cos\alpha} \int_0^{\Large\frac{\pi}{2}}\,\frac{\frac{1}{2}\,\sec^2\,\frac{\phi}{2}}{\left[\,\left(\dfrac{2\,\cos^2\,\frac{\alpha}{2}}{2\,\sin^2\,\frac{\alpha}{2}}\right) + \tan^2\,\frac{\phi}{2} \right]} \,d\phi \\ &=-\frac{2\left(2\,\sin\,\frac{\alpha}{2}\,\cos\,\frac{\alpha}{2}\right)}{2\,\sin^2\,\frac{\alpha}{2}}\,\int_0^{\Large\frac{\pi}{2}}\,\frac{1}{\left[\left(\dfrac{\cos \frac{\alpha}{2}}{\sin\,\frac{\alpha}{2}}\right)^2\,+\,\tan^2\,\frac{\phi}{2}\,\right]}\,d\left(\tan\,\frac{\phi}{2}\right)\\ &=-2\cot \frac{\alpha}{2}\,\int_0^{\Large\frac{\pi}{2}}\,\frac{1}{\left[\,\cot^2\,\frac{\alpha}{2} + \tan^2\,\frac{\phi}{2}\,\right]}\,d\left(\tan \frac{\phi}{2}\right)\,\\ &=-2\,\left.\tan^{-1} \left(\tan \frac{\alpha}{2} \tan \frac{\phi}{2} \right) \right|_0^{\Large\frac{\pi}{2}}\\ &=-\alpha \end{align}
Therefore:
$$I(\alpha) = C - \frac{\alpha^2}{2}$$
However by definition, $I\left(\frac{\pi}{2}\right) = 0$, hence $C = \dfrac{\pi^2}{8}$ and
$$I(\alpha) = \frac{\pi^2}{8}-\frac{\alpha^2}{2}.$$
The integral we want to evaluate is