L'Hôpital's Rule and Improper Integrals

Prerequisite: calculus

Improper integrals are definite integrals of infinite partitions. Meaning, the interval of integration stretches on to infinity but still has a definite area. Whuuut? Yep. Provided that the infinite end approaches a limit. You will see~

Due to the infinite nature of improper integrals, you need L'Hôpital's rule to further deal with limits. It is important when you run into results of (0 / 0), (∞ / ∞), ∞•0, ∞-∞, and the such, which are indeterminate forms. The rule to finding their limits is this:

[lim x-->a] f(x) / g(x)   =   [lim x-->a] f'(x) / g'(x)

provided that f(x) / g(x) is an indeterminate form.

And sometimes, you may have to derive many times over until you get to an actual number, or a dead end indicating that the limit does not exist.

As a side note, it is useful to get rid of logarithms that lead to 1^∞, 0^0, ∞^0, and the like. For example:

[lim x-->a] ln(f(x)) = L   becomes   [lim x-->a] f(x) = e^L

Next, you should know about the comparison test. If a function converges, then it approaches a limit and has a definite integral. On the other hand, a function that diverges does not approach a limit and does not have a finite integral.

The comparison test is similar to the sandwich theorem. Consider functions f(x) and g(x) that are continuous on [a, +∞) and 0 ≤ f(x) ≤ g(x):

1)  ∫ f(x) dx [a,+∞) converges if ∫ g(x) dx [a,+∞) converges.

2)  ∫ g(x) dx [a,+∞) diverges if ∫ f(x) dx [a,+∞) diverges.

It is fairly intuitive. Nothing crazy.

Now for solving improper integrals. Say, you want to take the integral of f(x) over interval [a,+∞). This is what you do:

Set up.

∫ f(x) dx   [a,+∞)

Antidifferentiate.

f(x) --> F(x)   [a, +∞)

Deploy integration evaluation theorem.

∫ f(x) dx   [a , +∞)    =   F(+∞) - F(a)

Find limit of F(+∞).

= [ lim(x --> +∞)   F(x) ] - F(a)

If the limit diverges, then the integral is not finite. If the limit does exists, evaluate the total to get the integral. There~

Say, you want to take the integral of f(x) over the interval (-∞, +∞). Just split the integral at 0.

∫ f(x) dx  (-∞, +∞)

= ∫ f(x) dx  (-∞, 0]   +    ∫ f(x) dx  [0, +∞)

Then do the same~