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Integration with partial fractions

integration with partial fractions

Integration of rational functions is often made considerably easier by decomposing them into partial fractions.

Observe that an expression of the form $$ \frac{A}{x+b} $$ Integrates to give $$ \int \frac{A}{x+b}dx = A\ln(x+b) + c $$

Consequently, decomposition of $\frac{P(x)}{Q(x)}$ into an expression of the form $$ \frac{P(x)}{Q(x)} = L(x) + \frac{A_1}{x+b_1} + \dots +\frac{A_n}{x + b_n} $$ where $L(x)$ is a polynomial determined by long division and the remaining terms are determined through a partial fractions decomposition, gives an easily integrated expression. There are some issues if the partial fractions decomposition doesn't return terms that all have linear denominators, but this is addressed in the further maths syllabus. 

integration with partial fractions 1

Evaluate the integral below. $$ \int \frac{12}{x^2 +5x +6}dx $$
solution - press button to display
Decomposing the integrand using partial fractions gives $$ \frac{12}{x^2+5x+6} = \frac{A}{x+3} + \frac{B}{x+2} $$ To determine $A$ and $B$, cross multiply and substitute $x=-2$ and $x=-3$ $$ 12 = A(x+2) + B(x+3) $$ Substitution of $x=-2$ yields $B = 12$, substitution of $x=-3$ yields $A=-12$ Our integrand is therefore $$ \frac{12}{x^2+5x+6} = \frac{-12}{x+3} + \frac{12}{x+2} $$ Integration proceeds follows $$ \begin{align} \int \frac{12}{x^2+5x+6}dx &= \int \frac{-12}{x+3} + \frac{12}{x+2} dx \\ &=-12\ln(x+3) + 12\ln(x+2) + c \\ &= 12\ln\left(\frac{x+2}{x+3}\right) + c \end{align} $$