I am trying to find the moment generating function for a random variable with the probability distribution of $$ f_X(x)= \begin{cases} \frac{1}{2π}\sqrt{4-x^2},& -2\le x\le 2 \\ \\ 0, &\text{ otherwise} \end{cases} $$
For the moment generating function I have $$ M(t) = E[e^{tx}]=\int_{-2}^{2}e^{tx}\frac{1}{2π}\sqrt{4-x^2}\ dx\\ =\frac{1}{2π}\int_{-2}^{2}e^{tx}\sqrt{4-x^2}\ dx $$
But I am having trouble trying to integrate this.
I have tried to find the expectation $E[X]$, $E[X^2]$, ... individually and it turns out the expectation is equal to $0$ when $n$ is even for the $n^{th}$ moment, so I am assuming the integral would contain a trigonometric function in it. Can somebody help me with this? Thank you.
$$M(t) = E[e^{tx}]= \frac{1}{2π}\int_{-2}^{2}e^{tx}\sqrt{4-x^2}\ dx=\frac 2 \pi\int_{-1}^{1} e^{2 t y}\sqrt{1-y^2}=\frac{I_1(2 t)}{t}$$ where appears the modified Bessel function of the first kind.