## question on integration of $f(x)\delta(g(x))$

I am learning integration with dirac delta. I do not understand this result by Maple When I work it by hand, using known relation (here is an image of the page) Therefore

$$\int_{0}^{\pi}f\left( x\right) \delta\left( g\left( x\right) \right) dx=\sum_{x_{0}}\frac{f\left( x_{0}\right) }{\left\vert g^{\prime}\left( x_{0}\right) \right\vert }$$

Where the sum is over all zeros of $$g\left( x\right)$$ in the integration interval, which is $$\left\{ 0,\pi\right\}$$ since $$\sin\left( x\right)$$ is zero at these points, and since $$g^{\prime}\left( x\right) =\cos\left( x\right)$$, and $$f\left( x\right) =\frac{1}{x}$$, then I get

\begin{align*} \int_{0}^{\pi}f\left( x\right) \delta\left( g\left( x\right) \right) dx & =\lim_{x\rightarrow0}\frac{\frac{1}{x}}{\left\vert \cos\left( x\right) \right\vert }+\lim_{x\rightarrow\pi}\frac{\frac{1}{x}}{\left\vert \cos\left( x\right) \right\vert }\ & =\infty \end{align*}

So I am doing something wrong or not using the above result correctly. Is it ok to use the zeros of $$g(x)$$ if they located at the edge of the interval as in this case? There are no other zeros inside the interval.

Does this mean if there are no zeros of $$g(x)$$ in the interval of integration, the integral is therefore zero, since nothing to sum?

How did Maple obtain zero for the above?

Posted on Categories proxies