6D1MoreExamples

Integration using Partial Fractions: More Examples

Example 1

math . \qquad \text{Find } \; \displaystyle{\int} \dfrac {4}{x^2-4x-5} \; dx \qquad x \neq \lbrace 5, \; -1 \rbrace math

First separate using partial fractions

math \\ . \qquad \dfrac {4}{(x-5)(x+1)} = \dfrac{A}{x-5} + \dfrac{B}{x+1} \\. \\ . \qquad \dfrac {4}{(x-5)(x+1)} = \dfrac{A(x+1)+B(x-5)}{(x-5)(x+1)} math . math . \qquad 4=A(x+1)+B(x-5) math . math \\ . \qquad \text{Let } \; x=-1 \; \Rightarrow \; 4=A(0)+B(-6) \; \Rightarrow \; B = - \dfrac{2}{3} \\. \\ . \qquad \text{Let } \; x=5 \; \Rightarrow \; 4=A(6)+B(0) \; \Rightarrow \; A = \dfrac{2}{3} math . math . \qquad \text{Thus } \; \dfrac {4}{(x-5)(x+1)} = \dfrac{2}{3} \left( \dfrac{1}{x-5} \right) - \dfrac{2}{3} \left( \dfrac{1}{x+1} \right) math

Now do the integral

math . \qquad \displaystyle{\int} \dfrac {4}{x^2-4x-5} \; dx \\. \\ . \qquad = \displaystyle{\int} \dfrac{2}{3} \left( \dfrac{1}{x-5} \right) \; dx - \displaystyle{\int} \dfrac{2}{3} \left( \dfrac{1}{x+1} \right) \; dx math . math . \qquad = \dfrac{2}{3} \log_e \big| x-5 \big| - \dfrac{2}{3} \log_e \big| x+1 \big| + c math . math . \qquad = \dfrac{2}{3} \big( \log_e \big| x-5 \big| - \log_e \big| x+1 \big| \big) + c math . math . \qquad = \dfrac{2}{3} \log_e \dfrac {|x-5|}{|x+1|} + c \qquad x \neq \lbrace 5, \; -1 \rbrace math

Example 2

math . \qquad \displaystyle{\int} \dfrac{3x^2+3x+4}{(x-2)(x-3)} \; dx \qquad x \neq \lbrace 2, \; 3 \rbrace math

Notice the numerator has degree equal to denominator so first do long division (you may be asked to show working)

math \\ . \qquad \dfrac{3x^2+3x+4}{(x-2)(x-3)} \\. \\ . \qquad = 3 + \dfrac {18x-14}{(x-2)(x-3)} math

Now separate the rational expression into partial fractions (you __may__ be asked to show working)

math . \qquad = 3 - \dfrac {22}{x-2} + \dfrac{40}{x-3} math

Now do the integral

math \\ . \qquad \displaystyle{\int} \dfrac{3x^2+3x+4}{(x-2)(x-3)} \; dx \\. \\ . \qquad = \displaystyle{\int} 3 - \dfrac {22}{x-2} + \dfrac{40}{x-3} \; dx math . math \\ . \qquad = 3x - 22\log_e \big| x-2 \big| + 40\log_e \big| x-3 \big| + c \\. \\ . \qquad = 3x - \log_e \big| x-2 \big|^{22} + \log_e \big| x-3 \big|^{40} + c \\. \\ . \qquad = 3x + \log_e \dfrac {|x-3|^{40}}{|x-2|^{22}} + c \qquad x \neq \lbrace 2, \; 3 \rbrace math

Example 3

math . \qquad \displaystyle{\int} \dfrac{2x-1}{(x+3)^2} \; dx \qquad x \neq -3 math

Separate using partial fractions. Note the two denominators because of the pefect square (you __may__ be asked to show working)

math \\ . \qquad \displaystyle{\int} \dfrac{2x-1}{(x+3)^2} \; dx \\. \\ . \qquad = \displaystyle{\int} \dfrac{-2}{x+3} \; dx + \displaystyle{\int} \dfrac {5}{(x+3)^2} \; dx math . math \\ . \qquad = -2\log_e \big| x+3 \big| +5(-1)(x+3)^{-1} + c \\. \\ . \qquad = \log_e \left( \dfrac{1}{|x+3|^2} \right) - \dfrac{5}{x+3} + c \qquad x \neq -3 math

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