3.5.6—Integration by substitution

Set u=g(x) when g′(x) appears, replace dx and every occurrence of x, integrate in u, then substitute back. A definite integral can instead use transformed limits.

Do not mix x and u in the same unfinished integral. Choose a substitution that removes the inner function rather than merely renaming it.

∫x√(x²+1)dx with u=x²+1 gives ½∫u^{1/2}du=(1/3)(x²+1)^{3/2}+C.

Substitution is incomplete if the differential factor is missing; the presence of g(x) alone is not enough.