Consider the indefinite integral ∫5x3+2x2−42x−21x2−9 dx\displaystyle \int{\frac{{{5}{x}^{{3}}+{2}{x}^{{2}}-{42}{x}-{21}}}{{{x}^{{2}}-{9}}}}\ {\left.{d}{x}\right.}

Then the integrand decomposes into the form
ax+b+cx−3+dx+3\displaystyle {a}{x}+{b}+{\frac{{{c}}}{{{x}-{3}}}}+{\frac{{{d}}}{{{x}+{3}}}}
where
a\displaystyle {a} =
b\displaystyle {b} =
c\displaystyle {c} =
d\displaystyle {d} =

Integrating term by term, we obtain that
∫5x3+2x2−42x−21x2−9 dx=\displaystyle \int{\frac{{{5}{x}^{{3}}+{2}{x}^{{2}}-{42}{x}-{21}}}{{{x}^{{2}}-{9}}}}\ {\left.{d}{x}\right.}=
  +C\displaystyle +{C}

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