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Practice question simplifying fractions for integration by using partial fractions.

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Simplify the following expression using partial fractions and find the general solution.

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$\\frac{\\var{a}x\\var{b}}{(x-\\var{c})(x+\\var{d})}$

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The following must be decomposed by partial fractions to be integrated

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$\\frac{\\var{a}*x\\var{b}}{(x-\\var{c})(x+\\var{d})}$

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The denominator contains two linear factors, $(x-\\var{c})$ and $(x+\\var{d})$.

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Thus, the decomposition of the original expression is as follows

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$\\frac{\\var{a}*x\\var{b}}{(x-\\var{c})(x+\\var{d})} = \\frac{A}{x-\\var{c}} + \\frac{B}{x+\\var{d}}$

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As both factors are linear, the cover-up rule can be used:

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$A\\lim=\\frac{\\var{a}*\\var{c}\\var{b}}{(\\var{c}+\\var{d})}$ = \\[\\var{((a*c)+b)/((c)+d)}\\]

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$B\\lim=\\frac{\\var{a}*\\var{d}\\var{b}}{(\\var{d}-\\var{c})}$ = \\[\\var{((a*(-d))+b)/((-d)-c)}\\]

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Sub back into the partial fractions

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\\[\\frac{\\var{((a*c)+b)/((c)+d)}}{x-\\var{c}} + \\frac{\\var{((a*(-d))+b)/((-d)-c)}}{x+\\var{d}}\\]

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The expressions can now be integrated

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\\[\\int{\\frac{\\var{((a*c)+b)/((c)+d)}}{x-\\var{c}} + \\frac{\\var{((a*(-d))+b)/((-d)-c)}}{x+\\var{d}}}.dx\\]

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Integration by parts

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\\[\\int{\\frac{\\var{((a*c)+b)/((c)+d)}}{x-\\var{c}}}.dx + \\int{\\frac{\\var{((a*(-d))+b)/((-d)-c)}}{x+\\var{d}}}.dx\\]

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Move the constants outside

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\\[\\var{((a*c)+b)/((c)+d)}\\int{\\frac{1}{x-\\var{c}}}.dx + \\var{((a*(-d))+b)/((-d)-c)}\\int{\\frac{1}{x+\\var{d}}}.dx\\]

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Substitution

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let $u=x-\\var{c}$  therefore $\\frac{du}{dx}=1 \\therefore dx=du$

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let $v=x+\\var{d}$  therefore $\\frac{dv}{dx}=1 \\therefore dx=dv$

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Sub these substitutions in to the integrals

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\\[\\var{((a*c)+b)/((c)+d)}\\int{\\frac{1}{u}}.du + \\var{((a*(-d))+b)/((-d)-c)}\\int{\\frac{1}{v}}.dv\\]

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Integration then yields

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\\[ \\var{((a*c)+b)/((c)+d)} ln(u) + \\var{((a*(-d))+b)/((-d)-c)}ln(v) +C \\]

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Sub back in the original denominators for the general solution

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\\[ \\var{((a*c)+b)/((c)+d)} ln(x-\\var{c}) + \\var{((a*(-d))+b)/((-d)-c)}ln(x+\\var{d}) +C \\]

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Decompose the fraction into partial fractions and state them as follows. The constants do not need to be evaluated yet.

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$\\frac{\\var{a}x\\var{b}}{(x-\\var{c})(x+\\var{d})}$=[[0]]

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Evaluate the constants A and B in decimal form.

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$A =$ [[0]] and $B =$[[1]]

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State the partial fractions using the constants A and B. Add the constants as decimals.

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$\\frac{\\var{a}x\\var{b}}{(x-\\var{c})(x+\\var{d})}$=[[0]]

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Integrate the partial fractions with respect to dx to find the general solution.

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\\[\\int{\\frac{\\var{((a*c)+b)/((c)+d)}}{x-\\var{c}}}.dx + \\int{\\frac{\\var{((a*(-d))+b)/((-d)-c)}}{x+\\var{d}}}.dx\\]

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Use C as the constant of integration.

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$y=$[[0]]

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