How do you solve hard integration questions using information you know

lets try a few integrations. what is integral of dx/(1+x)? You can see that it is in the form of f'(x)/f(x) so it is ln[f(x)] right? now what about integral of (e^x/(1+e^x))dx that is also in the form of f'(x)/f(x) right? so the answer follows and what would that be? ln(e^x +1)+c*. Now this is true and it follows but another way to think about it is that de^x/dx = e^x so if you have e^xdx we can write de^x in its place and that works too. lets see it in practice. integral of 1/(1+x^2)dx = arctan(x)+c, so if we have integral of e^x/(1+e^2x)dx = integral 1/(1+(e^x)^2)de^x = arctan(e^x)+c [that is we assume x=e^x]. can you see how it is used in integral of cos(x)(sin(x))^ndx? how do you think we will go about integral of 1/(1+e^x)dx?