What are BesselJ functions? I solved an integration on mathematica which gives BesselJ functions and some other terms. I explored mathematica help and google but could not understand the difference between different types of bessel functions. Specially BesselJ function and its explicit form is my target. 

$~\quad~$ Bessel and Struve functions also appear in the following context: What happens when 
we evaluate $($definite$)$ integrals of the form $\displaystyle\int_0^\lambda f\Big(g(x)\Big)~dx$, where $\big\{f,g\big\}\in\big\{\sin,~\sinh,$ 
$\cos,~\cosh\big\},~$ and $\lambda$ is either $\dfrac\pi2$ or $\infty$, depending on whether g is either a trigonometric or 
a hyperbolic function. Thus, for $a>0$ we have the following identities:


$$
\int_0^\tfrac\pi2~\sin~\big(a~\sin x\big)~dx~=~\int_0^\tfrac\pi2~\sin~\big(a~\cos x\big)~dx~=~\frac\pi2~H_0(a)
\\
\int_0^\tfrac\pi2 \sinh\big(a~\sin x\big)~dx~=~\int_0^\tfrac\pi2\sinh\big(a~\cos x\big)~dx~=~\frac\pi2~L_0(a)
$$



$$
\int_0^\tfrac\pi2~\cos~\big(a~\sin x\big)~dx~=~\int_0^\tfrac\pi2~\cos~\big(a~\cos x\big)~dx~=~\frac\pi2~J_0(a)
\\
\int_0^\tfrac\pi2 \cosh\big(a~\sin x\big)~dx~=~\int_0^\tfrac\pi2\cosh\big(a~\cos x\big)~dx~=~\frac\pi2~I_0(a)
$$



$$\begin{align}
\int_0^\infty\sin(a~\sinh x)~dx~&=~\quad\dfrac\pi2~\Big(I_0(a)-L_0(a)\Big)
\\
\int_0^\infty\sin(a~\cosh x)~dx~&=~\quad\dfrac\pi2\cdot J_0(a)
\\
\int_0^\infty\cos(a~\sinh x)~dx~&=~\quad~\quad~K_0(a)
\\
\int_0^\infty\cos(a~\cosh x)~dx~&=~-\dfrac\pi2\cdot Y_0(a)
\end{align}$$

