Sunday, 28 July 2013

integration - How to solve the following integral intfracpi20sqrt[3]sin8xcos4xdx?



How to solve the following integral?



π203sin8xcos4xdx



Preferably without the universal substitution sin(t)=2tan(t/2)1+tan2(t/2)


Answer



Using B(a,b)=2π/20sin2a1xcos2b1xdx, your integral is12B(116,76)=Γ(116)Γ(76)2Γ(3)=5144Γ(56)Γ(16)=5π144cscπ6=5π72.Here the first = uses B(a,b)=Γ(a)Γ(b)Γ(a+b), the second Γ(a+1)=aΓ(a), the third Γ(a)Γ(1a)=πcscπa.


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