Quick description
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Which of the following descriptions best fits the problem you are trying to solve?
I have an indefinite integral and I want to evaluate it. In that case, you have some function and you would like to find a function
that differentiates to
. There is no systematic way of doing this, but there are plenty of tricks, such as integration by substitution and integration by parts. For plenty of integration tricks, try the Tricki page on evaluating indefinite integrals.
I have a definite integral and I want to evaluate it.
, probably your best bet is to evaluate the corresponding indefinite integral
, using the advice about evaluating indefinite integrals, to find a function
that differentiates to
, in which case the answer is
. Another technique that sometimes works is differentiation under the integral sign: you differentiate the integral with respect to some variable, obtain a different quantity that is easier to integrate, and integrate it to get your answer. And another is to use an appropriate series expansion: you write
as
, integrate the
, and add up the results.
, the best method may be to find a function
that differentiates to
and to take the answer
, assuming that you can make good sense of
by an appropriate limiting argument. However, you have other options as well, such as contour integration and differentiation under the integral sign.
I have a definite integral that looks impossible to evaluate exactly in terms of standard functions, but I want to approximate it.
I have an integral over an interval of infinite length and I want to show that the integral is finite.
is finite as well.
such that
for every
and such that you can prove that
is finite. There are many ways of implementing this basic idea: see bound the integrand by something simpler for details.
is not finite, but
oscillates enough for the resulting cancellation to cause
to be finite.
, which is finite because it oscillates faster and faster, while remaining bounded, so the function
makes smaller and smaller oscillations and converges to a limit. See methods for bounding oscillatory integrals for details about how to handle this kind of situation.
I have an integral over an interval of finite length and I want to show that it is finite.
is bounded.
is not finite, but a positive blow-up on one side of a point is sort of compensated for by a negative blow-up on the other side.
. Strictly speaking, this is neither Riemann integrable (because it is unbounded) nor Lebesgue integrable (because it is not absolutely integrable). However, the limit of
exists (and is equal to
). This is an improper integral. When it is less obvious that the limit exists, you may need to use tools for estimating integrals. An example would be the function
, say, which is also improperly integrable between
and
. Improper integrals are important in Fourier analysis, as they arise in the definition of the Hilbert transform.
I need to prove that a function is Riemann integrable.
exists. To determine this, you will need to think about methods for estimating integrals.
-type discontinuities, then your best bet is to forget the precise definition of the function and prove the general result that a bounded function that is continuous except at finitely many points is Riemann integrable. This kind of proof is discussed below.
- The advice to give here depends very much on what you are allowed to assume. For instance, often it is a good idea to use the theorem that a continuous function on a closed bounded interval is Riemann integrable. But if that is precisely the result you want to know how to prove, then obviously you can't use it. So let us get a feel for what exactly your problem is.
- I have just been given the definition of the Riemann integral and want to prove from first principles that
is Riemann integrable.
- Almost always, the best approach is to use Riemann's criterion: a function
defined on the interval
is Riemann integrable if and only if the following statement holds. For every
there exists a dissection
such that
, where
is the upper sum associated with
and the dissection, and
is the lower sum. For more details, see how to prove from first principles that a function is Riemann integrable.
- Almost always, the best approach is to use Riemann's criterion: a function
- I have just been given the definition of the Riemann integral and want to prove from first principles that
- I am happy to assume a few basic facts about Riemann integrable functions.
- Three basic facts, or collections of facts, you might use are these. First, a continuous function defined on a closed bounded interval is Riemann integrable. Secondly, a bounded monotone function defined on a bounded interval is Riemann integrable. Thirdly, if
and
are integrable on
, then you can make other integrable functions such as linear combinations
, the pointwise product
, and so on.
If these results are not enough to solve your problem, then there is a very useful principle that may be. It's that bounded functions that behave almost everywhere are Riemann integrable. This, for example, can be used to show that the functiondefined on
that equals
when
and
when
is Riemann integrable: this function is bounded, and it is continuous everywhere except at
.
- Three basic facts, or collections of facts, you might use are these. First, a continuous function defined on a closed bounded interval is Riemann integrable. Secondly, a bounded monotone function defined on a bounded interval is Riemann integrable. Thirdly, if
I have a function that I am told is Riemann integrable, and I want to prove something about
.
be a Riemann integrable function and suppose that
for every
. Prove that
.