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,a≥0
Answer
The given integral is not converging, so I assume you wanted to study:
f(a)=∫+∞01−cos(ax)x2dx
that is an even function, hence we can assume a≥0 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:
∀r∈R,∫+∞01−cos(rx)xdx=π2|r|.
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