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Which integrals are simpler to integrate

Quick description

An integral may be too complicated to evaluate completely by the exact methods of freshman calculus, such as substitution or integration by parts. Nevertheless, it is still worthwhile to use these methods as much as possible to simplify the integral to a more manageable form.

Note that "simpler" may not mean "cleaner" or "shorter". Sometimes, one would rather deal with a messy sum of ten easily controllable integrals than one short but difficult integral.

Prerequisites

Basic real analysis; calculus

General discussion

In order to simplify an integral, one needs a good sense as to what integrands are going to be easier for you to integrate; this method will not be so productive if you manage to bound a relatively tractable expression f(x) by another expression M(x) which is in fact harder to integrate than the original.

It also makes a difference what your ultimate aim is, and in particular whether one wants to evaluate an integral exactly, or approximately. Some integrands are easy to evaluate approximately, but difficult to compute exactly; conversely, some integrals can be computed exactly via some trickery, but are not obviously easy to approximate accurately.

With enough practice, one will be able to quickly evaluate what integrands are preferable to end up with. But we can list here some general examples:

Good integrands for exact integration

  • Polynomials and constants.

  • Trigonometric polynomials.

  • Sums of simpler expressions (since one can simply integrate each sub-expression separately).

  • Exponentials of linear functions.

Bad integrands for exact integration

  • Integrands with complicated denominators. Here, one should seek substitutions or partial fraction decompositions to simplify the expression.

  • Exponentials or trigonometric functions of expressions more complicated than a linear function. Here, a substitution to linearize the exponent can be useful.

Good integrands for approximate integration

  • Integrands which can be integrated exactly and explicitly, e.g. x^a ,e^{tx}, \sin(x), etc. This is clear; but note that these are not the only type of integrands that one would be happy to end up in. In particular one could be perfectly willing to end up with an integrand which is algebraically messy (and so difficult to integrate exactly), but analytically pleasant (easy to bound).

  • Integrands which do not depend too much on the variable of integration. For instance, x e^{\sin(y^6)}/y^4 is very pleasant to integrate in x, but terrible to integrate in y. More generally, it is worthwhile to find bounds which reduce the dependence on the integration variable, even if it makes the integrand look messier in other ways.

  • Integrands which do not depend on an unknown or complicated function. Suppose one had a complicated multiple integral involving some function f(x) which one did not understand very well. Then it would often be advantageous to manipulate the integral so that at least one of the integrations did not involve that function f, as that integral is more likely to be computable or controllable explicitly. This method works particularly well with "interchanging integrals or sums" or the Cauchy-Schwarz inequality.

  • Integrands which are sums, averages, expectations, or integrals of simpler expressions. Indeed, one can then take the sum or average outside of the integral (using Fubini's theorem if necessary) and deal with the simpler integrals instead. (Of course, one still has to then compute the sum or average, but one should be able to deal with at least half of the task of estimating the integral this way.)

  • Integrands which are piecewise equal to simpler expressions. Minima and maxima \max(f,g), \min(f,g) fall into this category, or functions which are piecewise constant on, say, unit intervals. In this case a divide and conquer strategy works well.

  • Integrands which are tensor products. For instance, when computing a double integral \int \int f(x,y)\ dx dy, one would often like to replace the integrand f(x,y) with a tensor product g(x) h(y), as one can then factor the resulting two-dimensional integral \int\int g(x) h(y)\ dx dy into two one-dimensional integrals (\int g(x)\ dx) (\int h(y)\ dy). All other things being equal, one-dimensional integrals tend to be easier to integrate than higher-dimensional integrals. Thus, be on the lookout for bounds which "separate" the independent variables from each other.

  • Integrands which are lower order with respect to a low-regularity function. In some applications, notably when applying the energy method in PDE, one is integrating an expression involving derivatives of an unknown function, say u(x). In many cases, only a limited number of derivatives of u are under control, so it becomes useful to use techniques such as integration by parts to move derivatives off of the unknown function u onto smoother functions (e.g. cutoff functions), or alternatively to rebalance the distribution of derivatives on many factors of u.