Integration with partial fractions

Objective. Integrate decomposed rational functions logarithmically.

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Learn: the key idea

Integration with partial fractions uses decomposition to turn a complicated rational function into simple terms of the form A/(x−a), each integrating to A ln|x−a|+c. Example: ∫7/((x−1)(x+2)) dx first splits to (7/3)/(x−1)−(7/3)/(x+2), integrating to (7/3)ln|x−1|−(7/3)ln|x+2|+c. Repeated factors like B/(x−a)² integrate instead to −B/(x−a)+c, so recognise which rule applies to each term before integrating.

Worked example

Integrate the rational function 1/((x−1)(x+1)) using partial fractions.

1 $Decompose 1/((x-1)(x+1))=A/(x-1)+B/(x+1)$
2 $Find A=1/2 using x=1, and B=-1/2 using x=-1$
3 $Integrate each term: \int(1/2)/(x-1) dx - \int(1/2)/(x+1) dx$
4 $Combine results using logarithm laws$

Answer: (1/2)ln|x−1|−(1/2)ln|x+1|+c

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Common mistake. Students integrate B/(x−a)² as B ln|x−a| instead of −B/(x−a)+c, confusing it with the simple linear-factor rule; always check the power before choosing the integration result.

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