indefinite integration of sin(x)

[Iness]

Hi everyone, so for the integration of sin(x) from -infinity to +infinity i've followed the method below. Yet, after doing some researches in the internet i found that the integral of sin in this case does NOT converge. So can you help me fing what's wrong with my method ?

 

[Dr.Peterson]

All of your work assumes convergence! Check the theorems you are using, to see what their conditions are. But in general, if you don't yet know that something is a number at all, you can't manipulate it.

So your conclusion is really, "IF the integral converges, THEN it is 0".

 

[Steven G]

The first equal sign on the 3rd line is not valid. That makes everything that follows incorrect.

Also just because an integral converges you can't then say that implies that the integral = 0. And this is exactly what you wrote!!

 

[MarkFL]

While it is true that:

[MATH]\lim_{t\to\infty}\left(\int_{-t}^t \sin(x)\,dx\right)=0[/MATH]
This is less general than the integral posted.

We cannot assume that the lower and upper bound are moving away from the origin at the same rate.

 

[Steven G]

The first equal sign on the 3rd line is not valid. That makes everything that follows incorrect.

Also just because an integral converges you can't then say that implies that the integral = 0. And this is exactly what you wrote!!

Actually I was mistaken (I hope Subhotosh has the night off) about the equality sign being in correct. Sorry!