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What is the integral $$ \int_0^1 \exp(-a^2 (x-y)^2)) \, d x $$ According to mathematica it equals $$ \frac{\sqrt{\pi } (\text{erf}(a y)+\text{erf}(a(1- y)))}{2 a} $$

But how can this be proven? This must have been done somewhere, can you give me a reference? Thanks!

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    this follows directly from the definition of the error function found 100's of times on the net... – 2017-01-10
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    here you can find something about this integral http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf – 2017-01-10
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    @Dr. Trollhard Graubner this link is remarkably useless – 2017-01-10
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    why is this useless? – 2017-01-10
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    @Trollhard because it is only loosely (at best) related to the question – 2017-01-10

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Change variables to $u = a(x-y)$ and then use the definition of the error function. You will need to split up the integral and use the evenness of the the new integrand.