University Mathematics Handbook (2015)

V. Integral Calculus of Single-Variable Functions

Chapter 3. Improper Integral

Definite Integral is defined on finite interval , and function must be bounded in it.

An improper integral is defined on an infinite interval such as , , as well as on the entire straight line .

In addition, it is also defined on not necessarily bounded functions.

3.1  Integrals with Infinite Limits

a.  The improper integral of function defined on , and integrable on for all , exists or converges, if the limit exists and is finite, otherwise, we say the integral is not-convergent or divergent.

We write

b.  The improper integral on is

c.  Let be integrable in all closed interval , at any point . Let us define

If both of the integrals on the right are convergent, then we say the integrals on the left side are convergent. If at least one of the integrals on the right side is divergent, then the integral on the left is divergent. If the integral on the left side is divergent, then it is not dependent on the selection of .

d.  Convergence Tests

1.  Let be integrable on for all , when is a constant. Let and be two numbers.

If and for all in , then the integral is convergent.

If and for all in , then the integral is divergent.

2.  Cauchy Criterion for Convergence of an Improper Integral

Let function be integrable on for all . Then, integral is convergent if, and only if, for every there exist such that for all , there holds .

A similar proposition holds for integral .

3.  Comparison Test

Let and be two non-negative functions defined on , and integral on for every . If, for all , there holds , then, from the convergence of integral , there follows the convergence of integral .

If integral is divergent, then , is also divergent.

4.  Ratio Test

Let and be non-negative functions on . If the limit exists, then when:

a)  , the integrals and converge or diverge together. That is, of one integral is convergent, then the other is convergent, and if one of them is divergent, than the other is divergent.

b)  , then from the convergence of integral , there follows the convergence of integral .

c)  , then from the convergence of integral , there follows the convergence of integral .

5.  Abel Test

Let and be functions integrable on , for all . If the integral is convergent, and if is monotone and bounded on interval , then the integral is convergent.

6.  Dirichlet Test

Let and be functions integrable on , for all . Suppose there exists an such that for all , if is monotone on and . Then, the integral is convergent.

e.  Absolute Integrability

1.  Function is absolutely integrable on , if the improper integral is convergent.

We also say that integral is absolutely convergent.

2.  If is absolutely integrable on interval , then it is integrable on that interval. That is, if the integral is convergent, then integral is also convergent.

3.2  Unbound Function Integral

a.  Let defined and unbounded on semi-open interval , and, for all , the integral exists. If the limit exists, then is called the improper integral of and is denoted as .

b.  If is defined and unbounded on , and integrable on for all , let's define

c.  Let be:

1.  Defined on interval , except, possibly, at .

2.  Integrable on intervals and for all .

3.  Unbounded at the neighborhood of .

If the limits exist, then we say the improper integral of converges on , and its value is .

In case , the integral is called Cauchy principal value integral, and is denoted .

d.  Convergence Test

1.  Let be a functions defined on interval and integrable on for all , and let and be two real numbers. Then,

a)  If for all on interval , and if , then integral is convergent.

b)  If , for all on interval , and if , then integral is divergent.

2.  If the improper integral exists, then it is said that is absolutely integrable on .

3.  If is absolutely integrable on , then integral exists.

3.3  Gamma and Beta Functions

a.  Function is called gamma function.

b.  Gamma function properties:

1.  

2.  ,

3.  

4.   is continuous and has continuous differentials at .

5.  

c.  Function is called beta function.

d.  Beta function properties:

1.  

2.  

3.  

4.  

5.  

6.  

e.  The relation between the beta and gamma function