Does the Integral \(\int_0^\infty \sin(x) \, dx\) Have a Definite Value?

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In summary, Jo and the speaker discuss the existence of a definite value for the integral \int^{\infty}_0 sin(x)dx and the requirements for an improper integral of this type to exist. The speaker mentions that the original integral \int^{\infty}_0 sin(x) e^x dx comes from a step in their derivation for the solution to the IBVP of Burger's equation. Jo suggests that the integral may exist if a minus sign is added to the exponent. The speaker thanks Jo for their input.
  • #1
jollage
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Hi

Does this integral have a definite value [itex]\int^{\infty}_0 sin(x)dx[/itex]?

Jo
 
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  • #2
What is the requirement for an improper integral of this type to exist?
 
  • #3
arildno said:
What is the requirement for an improper integral of this type to exist?

Thanks. So you imply it's not a well-defined integral. Actually the original integral is [itex]\int^{\infty}_0 sin(x) e^x dx[/itex]. This original integral comes from a step during my derivation of the solution to the IBVP of Burger's equation. This original one seems more absurd, isn't it?

Jo
 
  • #4
Sure.
Most likely, you have missed a minus sign in the exponent, in which case the integral will exist.
 
  • #5
arildno said:
Sure.
Most likely, you have missed a minus sign in the exponent, in which case the integral will exist.

I see your point. Thanks.

Jo
 

FAQ: Does the Integral \(\int_0^\infty \sin(x) \, dx\) Have a Definite Value?

What does it mean for an integral to be undefined?

When an integral is undefined, it means that it does not have a finite value. This can occur when the function being integrated is not defined for certain values, or when the integral approaches infinity or negative infinity.

How can I determine if an integral is undefined?

To determine if an integral is undefined, you can look at the limits of integration and the function being integrated. If the function is undefined at any value within the limits, or if the integral approaches infinity or negative infinity, then the integral is undefined.

Can an integral be both defined and undefined?

No, an integral can only be either defined or undefined. If the function being integrated is well-defined and the limits of integration are within its domain, then the integral is defined. However, if there are any issues with the function or limits, the integral will be undefined.

How can I handle an undefined integral in my calculations?

If you encounter an undefined integral in your calculations, you may need to use special techniques such as integration by parts or partial fractions to evaluate it. Alternatively, you can also use software or online calculators that can handle undefined integrals.

Is an undefined integral the same as a divergent integral?

No, an undefined integral is not necessarily the same as a divergent integral. A divergent integral is one that does not have a finite value because it approaches infinity or negative infinity. On the other hand, an undefined integral may be due to other issues such as the function being undefined at certain points.

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