Is this possible to solve this definite integral?

Question

Is it possible to solve $$ \int_c^d e^{-ax-bx^2} dx $$ for $a$ and $b$ positive real numbers, and $c$ and $d$ real numbers?

I know it's possible when the bounds are infinity, but what about this case?

Thanks a lot.

Answer 1

With a basic knowledge of special functions, in particular the Error Function, we can easily find out that:

$$\int e^{-ax - bx^2}\ \text{d}x = \frac{\sqrt{\pi } e^{\frac{a^2}{4 b}} \text{erf}\left(\frac{a+2 b x}{2 \sqrt{b}}\right)}{2 \sqrt{b}}$$

Now you can fix the extrema as you like.

Special values

$$\text{erf}(0) = 0$$

$$\text{erf}(+\infty) = +1$$

$$\text{erf}(-\infty) = -1$$

More on Error Function: https://en.wikipedia.org/wiki/Error_function

Answer 2

HINT

Let $u$ such that $2bu=a$ then

$$(bx+u)^2=bx^2+2bux+u^2$$

and

$$\int_c^d e^{-ax-bx^2} dx=e^{u^2}\int_c^d e^{-(bx+u)^2} dx$$

then set $bx+u=y$ and use Error function.