As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students may know, most of us lose the chance to practice the calcaulation skills systematically after the calculus course. Moreover the most of the math books about the problem solving methods mainly discuss about the theory behind the problem solving (like the existence of the solution), not the general discussion on the actual way to get the exact result. Of course many students with the mathematical 'sense' or 'wit' may easily find the way to apply their mathematical common sense to disintegrate the complicated problems and break down into easy calcaulations, but someone like me without such bright eyes, it's not easy to just start to dig into the problem, like;
What trials should I give a chance to find this integral? $$\int \ln(2\sin(\frac{x}{2}))\cos(nx)ds$$ or,
What should I try first to find this sum? $$\sum_{n=1}^{\infty}\frac{(4n+2)(2n-2)!}{(2n-2)!!(2n+2)!!}$$
or
$$\prod_{n=1}^{\infty}(1+\frac{2ra}{n^2})$$ ?
So I first made a strategy to practice the skill by looking up the mathematical tables. I thought it is the best to be familiar with many specific cases to master the general calcuation skill. So I tried to find an aribtrary sum or integral or whatsoever that interests me. But no so long it tunred out to be futile, since the tables do not give me any hint or clue of the 'way' itself, which I eager to get to know, but only the 'result' like; $$\int_{0}^{\infty}\{(x+1)e^{-x}-e^{-\frac{x}{2}}\}\frac{dx}{x}=1-\ln2$$ Reading such entry of the integral table does not give me any clue of the way of finding the actual value. I realized that I really lack basic calculation skill so that I cannot use the integral table as an exercise workbook. Now I really feel lost in the flood of calculation. What might be the best way to increase the familiarity with a little bit complicated calculations?