I'm having trouble evaluating this integral
∫∞0e−ax2dx
My guess is that it would evaluate into something like
∫∞012e−ss12…dx=Γ(12)a122
When you do a substitution √s=√ax so that s=ax2. I'm having trouble convincing myself though that dds√s=(…a12) which would satisfy the answer that I provided.
Am I doing something wrong or is my guess wrong?
Answer
If you want to use the Gamma function the substitution is ax2=t, so “dx=12√at−1/2dt".
Then the integral appears as,
12√a∫∞0dtt−1/2e−t=12√aΓ(1/2) .
That's all.
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