Tuesday, September 19, 2017

calculus - Is there a novel way to integrate this without using complex numbers?



I've been reading a post on Quora about lesser known techniques of integration and I'm just curious if there's also a novel way to integrate this type of integral without resorting to complex analysis.




0cos(ax)dxx2,a0


Answer



The given integral is not converging, so I assume you wanted to study:



f(a)=+01cos(ax)x2dx
that is an even function, hence we can assume a0 WLOG. Integration by parts then gives:
f(a)=a+0sin(ax)xdx
and by replacing x with za we get:
f(a)=a+0sinxxdx=π2a
leading to:





rR,+01cos(rx)xdx=π2|r|.



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