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1.15 Integrating the partial fractions

Integrating the rational function ${f(x)\over g(x)}$ is therefore reduced to integrating a sum of a polynomial and some expressions of the form

(i)\quad{{\gamma}\over{(x-a)^{r}}}\quad{\hbox{or}}\quad(ii)\quad{{Hx+K}\over{(% x^{2}+2bx+c)^{r}}}.

Case (i) is straightforward to integrate as in Chapter 5 of MATH101: substituting $u=x-a$ we have ${du\over{dx}}=1$ and hence

\int{\gamma\over{x-a}}dx=\int{\gamma\,du\over u}=\gamma\log|u|+c=\gamma\log|x-% a|+c,

\int{\gamma\over{(x-a)^{r}}}dx=\int{\gamma\over u^{r}}du=-{\gamma\over{(r-1)u^% {r-1}}}+c=-{\gamma\over{(r-1)(x-a)^{r-1}}}+c.