How to integrate bessel function of order zero?

Question

How to do this integral:

$$ I = \int_{0}^{\infty} J_{0}(x)dx ?$$

Answer 1

We have $$L\{J_0(x):p\}=\frac{1}{\sqrt{1+p^2}} . . . . . (1)$$ (from Laplace transformation)

By the definition of Laplace transformation, $$L\{J_0(x):p\}=\int_0^{\infty} J_0(x) e^{-px} dx$$

So from $(1)$, $$\int_0^{\infty} J_0(x) e^{-px} dx=\frac{1}{\sqrt{1+p^2}}$$

Putting $p=0$ we have, $$\int_0^{\infty} J_0(x) dx=\frac{1}{\sqrt{1+0}}=1$$