Monday, 29 October 2018

calculus - Evaluating this integral using the Gamma function



I was wondering if the following integral is able to be evaluated using the Gamma Function.
0t12exp[a(t+t1)]dt


I already have a tedious solution that doesn't exceed the scope of the first few semesters of calculus, but I want to tackle this with the Gamma Function. I just don't know how or if it's even possible.




If anyone can give a hint, I'd really like to finish it on my own.



EDIT:
You are allowed to use the fact that
exp(x2)dx=π


Answer



Let t=u2, and the integral becomes




20duea(u2+1u2)=2e2a0duea(u+1u)2



Let v=u+1/u, then



u=12(v±v24)



du=12(1±vv24)dv



Now, it should be understood that as u traverses from 0 to , v traverses from down to a min of 2 (corresponding to u[0,1]), then from 2 back to (corresponding to u[1,)). Therefore the integral is




e2a2dv(1vv24)eav2+e2a2du(1+vv24)eav2



which is



2e2a2dvvv24eav2=e2a4dyy4eay=e2a0dqq1/2eaq



I guess the gamma function comes from this integral, but I find it easier to refer to gaussian integrals.


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