Is it valid to say that ∫ − ∞ ∞ f ( x ) d x = lim t → ∞ ∫ − t t f ( x ) d x

Question

Is it valid to say that       ∫          −      ∞        ∞    f  (  x  )  d  x  =      lim          t      →      ∞            ∫          −      t        t    f  (  x  )  d  x

Turynka2f · Accepted Answer

When it comes to the Cauchy Principle Value, the integral of an odd function across the real numbers is zero      ∫          −      ∞        ∞    o  (  x  )  d  x  =  0Where   o  (  x  ) represents an odd functionBut an improper integral across the real numbers does not exist if we cannot split it. In other words      ∫          −      ∞        ∞    f  (  x  )  d  xdoes not converge unless      ∫    0    ∞    f  (  x  )  d  x        ∫          −      ∞        0    f  (  x  )  d  xboth converge.

Is it valid to say that <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD">

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