Methods of Integration

Why integration is harder than differentiation. Differentiation has a constructive character — a derivative is, by definition, the limit

and there are systematic laws (sum, product, quotient, chain rule for composite functions) that let you differentiate any combination of known functions mechanically, step by step, with a guaranteed route to the answer. Asked to differentiate $\log x$, we know exactly how to proceed.

Integration has no such constructive procedure. Asked to find $\int \log x\,dx$, there is no ready-made mechanical rule to fall back on — not even where to start. Only a handful of general methods exist, each with restricted applicability, and — unlike differentiation — selecting the right method for a given integrand is itself a skill, acquired only through practice, since no single procedure works on every integrand.