Friday, May 20, 2016

calculus - What are other methods to evaluate int10sqrtlnxmathrmdx



10lnxdx
I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral.
y=lnx



ey=elnx=eln1x=1x




dx=dyey



10lnxdx=0y(dyey)=0eyy12dy=(12)!



(12)!=12π



10lnxdx=12π


Answer



As far as I know, there are essentially two ways for proving that Γ(12)=π.




The first way is to use integration by parts, leading to Γ(z+1)=zΓ(z), in order to relate Γ(12) to the Wallis product. The latter can be computed by exploiting the Weierstrass product for the sine function:
sinzz=+n=1(1z2π2n2)
by evaluating it in z=π2. The duplication formula for the Γ function follows from this approach.



The second way is to use some substitutions in order to relate Γ(12) to the gaussian integral
+ex2dx
that can be evaluated through Fubini's theorem and polar coordinates.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...