We've been learning the different methods of integration in a very artificial environment. We know that if we're in the "Integration by Substitution" section, we use substitution. If we're in the "Integration by Parts" section, we use integration by parts.
In the real world (by which we mean on exams), the directions probably won't say which method to use—we'll have to figure that ourselves.
The more we practice, the better we'll get at figuring out which method to use. We won't even have to think about it. In the meantime, we have some hints.
Use substitution when the integrand can be factored into something with an "inside function" u and something that's more-or-less the derivative of u (if the constant coefficients don't quite agree, that's ok).
We would use integration by substitution on
because x is a constant multiple of the derivative of x2, which is an inside function:
We can't use substitution on
If we try to let u = x2 it just doesn't work, because we have an extra factor of x hanging around:
Use integration by parts when the integrand factors into two things that both include the variable, but integration by substitution doesn't work!
We can use integration by parts on
because we can factor the x2 to get
and choose u = x and v' = xex2.
We wouldn't use integration by parts on
because this integral begs for substitution.
Use the partial fractions technique when you're asked to evaluate a rational function that
We can use the method of partial fractions on
because the numerator has degree 0, the denominator has degree 2, and the denominator factors into
x2 – 2x – 3 = (x – 3)(x + 1).
We wouldn't use the method of partial fractions on
because the denominator factors into
x2 + 2x + 1 = (x + 1)(x + 1).
These are not distinct linear factors.
Actually, it is possible to use the method of partial fractions on this example, but the setup is a bit more complicated. We'll stick to the simpler examples of integration by partial fractions.
Don't forget the first method we learned to find integrals: "thinking backwards." Sometimes you don't need substitution, parts, or partial fractions—you can simplify the integral and immediately see what to do with it.
We don't need anything fancy to find
Simplify the integral by squaring the integrand and then separating it out:
Then integrate each term:
Depending on how comfortable you are with thinking backwards, you might be able to do this one in your head:
However, you're still doing substitution behind the scenes, letting u = 2x + 3.
There's no reasonable way to think backwards from
That's what we learned integration by parts for.