# Solving an integral with square root in the exponent

I'm trying to solve to following integral (related to a previous post) $$\int_{-1}^{1}\exp{(Ax^2+Bx+C+(Dx+E)\sqrt{1-x^2})}dx$$ Any ideas on how to approach this?

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## 1 Answer

There is no known closed-formula for the reduced case $B=C=D=E=0$ so I highly doubt you can find any magic answer. If you are lucky enough you will potentially end up using special functions like incomplete Gamma, which is not very easy to use.

If you are comfortable with approaching this integral numerically, I suggest using adaptive Gaussian quadrature with a sufficient number of roots.

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I don't get it. In your reduced case $\int_{-1}^{1}\exp(Ax^2) dx=2\exp(Ax^2)$ – Wox Sep 17 '12 at 15:49
This is not correct. $\int_{-1}^1 \exp(Ax^2) dx$ cannot be solved explicitely because of the square. If an $x$ is put in front of the exponential then you can integrate, but there is no such thing here. – vanna Sep 17 '12 at 15:53
Sorry, copy&paste error. Maple gave this answer $\int_{-1}^{1}\exp(Ax^2)=\sqrt{\frac{-\pi}{A}}\text{erf}(\sqrt{-A})$ – Wox Sep 17 '12 at 15:59
Another solution – Wox Sep 17 '12 at 16:05
Both solutions you provided are not closed-formulae. Look at the definition of $\rm erf$ ;) – vanna Sep 17 '12 at 16:25