Evaluation of the integral $\int_0^1 e^{2t^2 -at} dt$

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I would like to integrate a function in the range $[0,1]$. I tried a lot of ways including Mathlab. All solutions come in terms of some error function.

I would like the answer in terms of $a$.

$$\int_0^1 e^{2t^2 -at} dt$$

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All solutions come in terms of some error function. I would like the answer in terms of a.

Since those error functions had a in their argument, you already got what you wanted.