Some antiderivatives of rational functions involve inverse trigonometric functions, and some involve logarithms. But inverse trig functions can be expressed in terms of complex logarithms. So is there a general formula for the antiderivative of any rational function that uses complex logarithms to unite the two concepts?

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Colettosryni · Accepted Answer

Write the rational function as  f  (  z  )  =                    p        (        z        )                    q        (        z        )              =                    p        (        z        )                              ∏                      j            =            1                    n                (        z        −                  r          j                )            where       r    j   are the roots of the denominator, and   p  (  z  ) is a polynomial. I&#039;ll assume   p has degree less than   n and the roots       r    j   are all distinct.Then the partial fraction decomposition of   f  (  z  ) is  f  (  z  )  =      ∑          j      =      1        n              p      (              r        j            )                      q        ′            (              r        j            )      (      z      −              r        j            )      where   p  (      r    j    )      /        q    ′    (      r    j    ) is the residue of   f  (  z  ) at       r    j  . An antiderivative is  ∫  f  (  z  )     d  z  =      ∑          j      =      1        n              p      (              r        j            )                      q        ′            (              r        j            )        log  ⁡  (  z  −      r    j    )

Shayla Osborne · Accepted Answer

If one uses partial fractions allowing complex numbers as coefficients, then the denominator of   p  (  x  )      /    q  (  x  ) factors as a constant times a product of terms of form   (  x  −      a    k        )    r   for a set of distinct complex       a    k  . Then partial fractions expresses   p  (  x  )      /    q  (  x  ) as the sum of a polynomial and terms of the form   c      /    (  x  −      a    k        )    j  , and so the antiderivative consists of that of the polynomial, and some logarithm terms from integrating any   c      /    (  x  −      a    k    ) terms , and some rational fractional terms coming from integrating   c      /    (  x  −      a    k        )    i   terms with   i  &amp;gt;  1. So it looks like all the terms are rational functions or log terms.

Some antiderivatives of rational functions involve inverse trigonometric functions, and

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