Proving function is improper riemann integrable

In summary, the discussion focuses on proving that the function f(x) = sin(x) / x is improper Riemann integrable without explicitly computing the integral. The attempted solutions involve finding an upperbound for f(x) and using the Cauchy criterion, with the suggestion to use integration by parts to show convergence.
  • #1
Pietjuh
76
0

Homework Statement



let f:[0,oo) -> R be given by f(x) = sin(x) / x for x>0 and f(0) = c. Prove that f is improper riemann integrable without computing the integral explicitly

The Attempt at a Solution



I've attempted to find a upperbound for f(x) such that the integral does not diverge. The most simple one is to use the fact that sin(x) <= 1 for all x, but this gives a divergent integral.

I've already proved that f is Lebesgue measurable for every c in R. So I could turn the integral [tex]\int_0^R f(x) dx[/tex] into a Lebesgue integral and then use one of the convergence theorems to try to show with them that the integral does not diverge. But I haven't succeded in doing this :(

Can anyone help me out?
 
Physics news on Phys.org
  • #2
As you've probably noticed, the integral of the absolute value of your function diverges. The fact an improper integral can be defined means you must be able to show that the positive parts can be combined with the negative parts to get a partial cancellation that can converge. Draw a graph and think about estimating. Does that help?
 
  • #3
Hoi Piet.

Use the Cauchy criterion!:
The improper integral [tex]\int_a^\infty f(x)dx[/tex] converges if and only if [itex]\forall \epsilon>0 \exists b>a[/itex] such that
[tex] c,d > b \Rightarrow \left|\int_c^d f(x)dx \right| < \epsilon[/tex]

----
You can use integration by parts to show that:
[tex]\left| \int_c^d \frac{\sin x}{x}dx\right|\leq 2\left(\frac{1}{c}+\frac{1}{d}\right)[/tex]
 
Last edited:

1. What is an improper Riemann integral?

An improper Riemann integral is a type of definite integral where one or both of the integration limits are infinite or the integrand is not defined at some points within the integration interval.

2. Why is it important to prove that a function is improper Riemann integrable?

Proving that a function is improper Riemann integrable ensures that the integral exists and can be evaluated, thus allowing for the application of the fundamental theorem of calculus and other important mathematical operations.

3. What are the criteria for a function to be improper Riemann integrable?

A function must meet two conditions to be improper Riemann integrable: 1) the function must have a finite limit as the integration limits approach infinity or a point of discontinuity, and 2) the integral of the absolute value of the function must converge.

4. How is the improper Riemann integral calculated?

The improper Riemann integral is calculated by splitting the integration interval into multiple intervals, each with a finite limit, and then taking the limit of the integral as the upper and lower limits approach these finite values.

5. Can any function be proven to be improper Riemann integrable?

No, not all functions can be proven to be improper Riemann integrable. For example, if a function has an infinite number of discontinuities within the integration interval, it cannot be proven to be improper Riemann integrable.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
268
  • Calculus and Beyond Homework Help
Replies
2
Views
781
  • Calculus and Beyond Homework Help
Replies
7
Views
886
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
5K
  • Calculus and Beyond Homework Help
Replies
1
Views
712
  • Calculus and Beyond Homework Help
Replies
9
Views
425
  • Calculus and Beyond Homework Help
Replies
2
Views
640
  • Calculus and Beyond Homework Help
Replies
2
Views
825
  • Calculus and Beyond Homework Help
Replies
7
Views
606
Back
Top