Calculus Small Changes, Big Ideas

Section 8.4 Partial fractions

• Application: logistic population growth

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• To integrate rational functions, we can use long division to separate out the polynomial part and the proper rational part, after which we can write every partial fraction as a sum of functions of the form

  \begin{equation*} \frac{A}{(x-\alpha)^{k}}\qand\frac{Bx+C}{(x^{2}+\beta x+\gamma)^{k}}\text{.} \end{equation*}

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• When we have \((x-\alpha)^{k}\) in the denominator, the partial fraction decomposition will contain an expression of the form

  \begin{equation*} \frac{A_{1}}{x-\alpha}+\frac{A_{2}}{(x-\alpha)^{2}}+\cdots+\frac{A_{k}}{(x-\alpha)^{k}}\text{.} \end{equation*}

  These can be integrated using the Anti-Power Rule or the natural logarithm.

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• When we have \((x+\beta x+\gamma)^{k}\) in the denominator, the partial fraction decomposition will contain an expression of the form

  \begin{equation*} \frac{B_{1}x+C_{1}}{x^{2}+\beta x+\gamma}+\frac{B_{2}x+C_{2}}{(x^{2}+\beta x+\gamma)^{2}}+\cdots+\frac{B_{k}x+C_{k}}{(x^{2}+\beta x+\gamma)^{k}}\text{.} \end{equation*}

  These can be integrated using (trigonometric) substitution.

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• Weierstrass substitution may fit here?

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