Simple Improper Integral Question (just a question of concept understanding)

In summary, an improper integral is an integral with either infinite limits of integration or an integrand that is undefined at one or more points within the interval of integration. It differs from a regular integral, which has finite limits and a well-defined integrand. An improper integral can have a finite value if the limits are finite and the integrand is well-behaved. To determine if an improper integral converges or diverges, the limit of the integral must be evaluated. Some common techniques for evaluating improper integrals include using the limit definition, splitting the integral, and using substitution or integration by parts.
  • #1
neden
18
0

Homework Statement



I am to determine whether the following integral is convergent or divergent

[tex]
\int_0^1 \frac{sin(x)}{x}
[/tex]

From what I hear since, lower limit is zero there is a removable discontinuity.
Thus just because of this, it is convergent? Can someone let me know if this
is correct.
 
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  • #2
Do you know that:

[tex]lim(x->0) \frac{sin(x)}{x}=1[/tex]

?

If yes, then you can see that the function is bounded on the interval and therefore integrable.
 
  • #3
Oh thanks!
 

1. What is an improper integral?

An improper integral is an integral where either the upper or lower limit of integration is infinite or the integrand is undefined at one or more points within the interval of integration.

2. How is an improper integral different from a regular integral?

A regular integral has finite limits of integration and the integrand is defined at all points within the interval of integration. An improper integral, on the other hand, may have infinite limits or undefined points within the interval, making it more challenging to evaluate.

3. Can an improper integral have a finite value?

Yes, an improper integral can have a finite value if both the upper and lower limits of integration are finite and the integrand is well-behaved within the interval.

4. How do you determine if an improper integral converges or diverges?

An improper integral converges if the limit of the integral as the upper or lower limit approaches infinity or an undefined point within the interval is a finite value. It diverges if the limit is infinite or does not exist.

5. What are some common techniques for evaluating improper integrals?

Some common techniques for evaluating improper integrals include using the limit definition of the integral, splitting the integral into two or more integrals with finite limits, and using substitution or integration by parts to transform the integral into a form that is easier to evaluate.

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