<s>[INST] How to calculate $\int _0^{\frac{1}{3}} \frac{e^{-x^2}}{\sqrt{1-x^2}}dx$ can someone give me a hint on how to calculate this integral?
$\int _0^{\frac{1}{3}} \frac{e^{-x^2}}{\sqrt{1-x^2}}dx$
Thanks so much!
 [/INST] $\int_0^\frac{1}{3}\dfrac{e^{-x^2}}{\sqrt{1-x^2}}~dx$
$=\int_0^{\sin^{-1}\frac{1}{3}}\dfrac{e^{-\sin^2x}}{\sqrt{1-\sin^2x}}~d(\sin x)$
$=\int_0^{\sin^{-1}\frac{1}{3}}e^\frac{\cos2x-1}{2}~dx$
$=e^{-\frac{1}{2}}\int_0^{2\sin^{-1}\frac{1}{3}}e^\frac{\cos x}{2}~d\left(\dfrac{x}{2}\right)$
$=\dfrac{e^{-\frac{1}{2}}}{2}\int_0^{2\sin^{-1}\frac{1}{3}}e^\frac{\cos x}{2}~dx$
Which can express in terms of Incomplete Bessel Functions
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