Comparing an integral and a summation 
If $I=\int_{-1}^{1} \frac{e^x}{\sqrt{1-x^2}} dx$ then the value of $S=\frac{1}{2^2}+\frac{1}{2^24^2}+\frac{1}{2^24^26^2}+\ldots$ is
A). $\frac{I-\pi}{\pi}$
B). $\frac{I-1}{\pi}$
C). $I-\pi$
D). $\frac{I-π}{2}$

I've tried applying Taylor series for the integral, and also tried doing the same after an Integration by parts, so as to get the π term,(although an unwanted e term appears as well), but couldn't make much progress with this.
I defined a function $f(x)=x^2+x^2(x/2)^2+x^2(x/2)^2(x/3)^2+\ldots$, and we need to evaluate the expression S, at $x=1/2$. I tried to form a relation with f(x),f'(x) and x, hoping that it would relate to I, but I hasn't able to form any relation.

Hint:
Note that $\exp(x) = \displaystyle \sum_{n \ge 0} \frac{x^n}{n!}$. Represent $\exp(x)$ in integral sign and continue calculations.
