Main Index Mathematical Analysis Real Analysis
  Subject Index
comment on the page

Improper integrals

Two essential restrictions are made when developing the Riemann’s integral typeset structure, both

should be bounded. In many applications these restrictions can be avoided. When either the integrand or the integration domain (or both) are unbounded the resulting integral is called improper. Thus we get two basic types of improper integrals:   

Improper integrals with unbounded integrands

Let typeset structureand function typeset structure be unbounded as typeset structure approaches typeset structure (from the left). The improper integral typeset structure is said to converge if the limit typeset structure exists and is finite. If this holds  we say that the value of the limit is the value of the improper integral. If the limit above does not exist or is improper we say that the improper integral diverges. An analogous definition can be formulated when the integrand is unbounded at the lower limit of the integration. Similarly, when the integrand has improper one-sided limits at an inner point typeset structure of interval typeset structure but the integrand is defined on typeset structure, then

∫ _ a^b f(x) d x = ∫ _ a^d f(x) d x + ∫ _ d^b f(x) d x

provided both improper integrals on the right-hand side exist.

Often it is difficult to find the corresponding antiderivative which is necessary to answer the divergence of convergence  of the given improper integral using the definition. As the terminology convergent or divergent indicates there are close connections between the theory of improper integrals and infinite series. As it was the case with infinite series also here the problem of convergence is easier to answer as the problem of finding the values to which the given improper integral converges. Various comparison arguments are of great help.

Comparison test: Let both functions typeset structure and typeset structure are unbounded at typeset structure and typeset structure for all typeset structure. Then

Improper integrals with infinite limits of integration

The improper integral typeset structure is said to converge if there exists the finite limit typeset structure. In this case we also say that the value of the limit is the value of the improper integral. If the limit above does not exist or is improper we say that the corresponding improper integral diverges.

If the improper integral  typeset structure is split into a sum of improper integrals (because typeset structure presents more than one improper behavior on typeset structure), then the integral converges if and only if any single improper integral is convergent.

Comparison test: Assume that  typeset structure for all typeset structure.Then

Convergence tests for improper integrals

Comparison with function of the type typeset structure with a real exponent typeset structure gives the following

typeset structure-test: the improper integral

Limit test:  Let typeset structure and typeset structure be two positive function defined on typeset structure. Assume that both functions exhibit an improper behavior at typeset structure and

lim _ (x -> a +) f(x)/g(x) = 1.

Then typeset structure is convergent if and only if typeset structure is convergent.

Absolute Convergence of Improper Integrals

The above tests of convergence for improper integrals are only valid for positive functions.  Since there is a very natural way of generating a positive function  from a given function just take its absolute value, consider a function typeset structure (not necessarily positive) defined on typeset structure and the positive function typeset structure  still defined on typeset structure. Both functions typeset structure and its absolute value typeset structure exhibit the same kind of improper behavior over typeset structure. Therefore, one may ask naturally what conclusion can be made if we know something about the integral

∫ _ a^b | f(x) | d x .

We have the following partial answer:

If the integral  typeset structure is convergent, then the integral typeset structure is also convergent.

This shows that the convergence of  typeset structure  carries more information than just convergence. Therefore we say that if the improper integral typeset structure converges that typeset structure is absolutely convergent. And if the improper integral typeset structure is convergent while the improper integral typeset structure is divergent, we say it is conditionally convergent.

Improper integrals and infinite series

There is a parallelism between improper integrals and infinite series. The integral test bridges the two notions.  One of its form states (for more details consult ) :

Integral test. Let typeset structure be a function defined for typeset structure, bounded, positive and monotonically decreasing to 0 as typeset structure. Let

S _ n = Underoverscript[∑, k = 1, arg3] f(k) ,         ... sp;        I _ n = Underoverscript[∫, 1, arg3] f(x) d x .

Then for typeset structure the difference typeset structure tends to a finite limit.

In particular, the series typeset structure is convergent if and only if the integral typeset structure exists and is finite.

Cite this web-page as:

Štefan Porubský: Improper Integral.

Page created  .