The general rule we said in class was to pick
u and
v'
so that the resulting integral was at least not harder than the
original integral. Some general rules that often make this happen:
- If the integrand is a product of a power of x and
another function, let u be the power of x, unless
this contradicts rule (2). Then let v' be the rest of
the integrand.
- If the integrand is a product with one function that you can
antidifferentiate and another that you can't, let u be
the function you can't antidifferentiate.
- If the integrand is only a single function that you can't
antidifferentiate, let u be that function and v'
be 1.
Let's look at an example or two:
int x sin(x) dx
is an example of rule (1).
int x ln(x) dx
is an example of rule (2).
int arcsin(x) dx
is an example of rule (3).