Monday, 20 January 2014

calculus - Integrals involving exponential functions and the gamma function





I'm having trouble evaluating this integral



0eax2dx



My guess is that it would evaluate into something like



012ess12dx=Γ(12)a122



When you do a substitution s=ax so that s=ax2. I'm having trouble convincing myself though that ddss=(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=12at1/2dt".
Then the integral appears as,
12a0dtt1/2et=12aΓ(1/2) .



That's all.


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