Example of integral pairs or triples ($I$, $J$, $K$...) A somewhat common trick when evaluating integrals is to add a second (or third) integral 
to help integrate the first. As an example if one where to compute 
$$
     I = \int \sin \log x \, \mathrm{d}x\,.
$$
A clever step would be to introduce the integral
$$
     J = \int \cos \log x \, \mathrm{d}x\,,
$$
and to turn the problem into a system of equations.
Since now it is easy to show by parts that
$$
\begin{align*}
I + J & = \left[ x \sin \log x - J \right] + J \\
J - I & = \left[ x \cos \log x + I \right] - I 
\end{align*}
$$
One can now solve this system of equation with ease. As one can see
to integrate $I$ or $J$ by itself is somewhat cumbersome (One can use the substitution $t = \log x$, and then rewrite the remaining expression into a complex one, or use more integration by parts, but both methods are somewhat tedious), but by looking at their products or difference the computation can be simplified. 
Another example are the integrals
$$
  I = \int \frac{\cos x}{\cos x + \sin x}\,\mathrm{d}x 
               \quad \text{and} \quad
  J = \int \frac{\sin x}{\cos x + \sin x}\,\mathrm{d}x 
$$
where the same technique can be applied. 
My question is: Are there any more examples of integrals that can be solved by transforming them into a system of equations?
(please do not say $e^x \cos x$, since this is the same as the first example)
Any suggestions,examples, reference to books, papers, is much appriciated. 

I'm not quite sure if this is what you want.
$$I=\int_{\frac{\pi}{3}}^{\frac{2}{3}\pi}\frac{\cos(x)\sin(3x)}{\cos\left(x+\frac{2}{3}\pi\right)}dx,\qquad J=\int_{\frac{\pi}{3}}^{\frac{2}{3}\pi}\frac{\cos(x)\sin(3x)}{\cos\left(x-\frac{2}{3}\pi\right)}dx.$$
Letting $t=\pi-x$, we can get $I=J.$ Hence, since we know $I=\frac{I+J}{2}$,  we get, setting $u=\cos x$,
$$I=\frac{I+J}{2}=\cdots =\int_{0}^{\frac 12}\frac{4u^2(4u^2-1)}{3-4u^2}du=\cdots=-\frac 76+\frac{\sqrt3}{2}\ln(2+\sqrt 3).$$
This is an example such that adding the two should make it easier to find though finding each integral seems difficult.
