# How to integrate bessel function of order zero?

How to do this integral:

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

• ...with one line of computer algebra... result: $0$. – David G. Stork Jun 7 at 3:15

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