How do I evaluate this integral using Beta functions?

Question

How do I evaluate this integral using Beta functions?

$$\int_a^\infty e^{2ax-x}dx$$

Edit 1:The answer given for this question is : $$e^{a^2}* \frac{\sqrt \pi}{2}$$

How do I get that using Beta functions? I have been breaking my head over it for a long time.

Edit 2: There's a possibility that the answer given is wrong. Even then is it possible to solve it using Beta functions?

Answer 1

Hint. One has $$ \int e^{cx}dx=\frac{e^{cx}}{c},\qquad c\ne0. $$

Answer 2

Integrate using $\int e^{ax}dx = \frac{1}{a}e^{ax}+c$.

$$ \begin{align*} \int e^{(2a-1)x}dx &= \left[ \frac{e^{(2a-1)x}}{2a-1} \right]^\infty_{a} \\ &= \frac{e^{a(2a-1)}}{1-2a}, \quad \textrm{for } \Re[a]<1/2 . \\ \end{align*} $$

Where we require $a$ be less than $1/2$ so that the integrand converges at $\infty$ and the denominator is not $0$.