I'm confused. Are you not taking (or have taken) a Calculus course? Generally speaking, you learn to use "substitution" in integration well before integrating trig functions. So the first thing you should think of when try to integrate is the substitution u= 5x so that du= 5 dx so that dx= (1/5)du. The integral becomes [tex]\int cos(u)(1/5)du[tex]. Because that "1/5" is a constant, we can take it outside the integral to get .
To integrate cos(5x^2), let u= 5x^2. Then du= 10xdx. And there's the rub! dx= 1/(10x) du so this would become . While we can take the constant, 1/10, out of the integral, we cannot take out the "1/x". IF the problem were THEN we could let so that du= 10xdx and xdx= (1/10)du so that the integral becomes . But without that "x" in the original integral, we cannot do that.
NO, there is no "general method to solve all of these". In fact, except for the very first, , none of the integrals can be written in terms of "elementary functions".