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How do you integrate # f(x) = (3x + 2) / [(x - 1)(x + 4)] # using partial fractions?

Answer 1

Christopher Agresta

Please see the following for your answer
c is a constant

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Answer 2

Aurora Alvarado

To integrate ( f(x) = \frac{3x + 2}{(x - 1)(x + 4)} ) using partial fractions, first decompose the rational function into partial fractions. Perform the partial fraction decomposition by expressing ( f(x) ) as the sum of two fractions with undetermined constants over the factors ( (x - 1) ) and ( (x + 4) ). The decomposition will be in the form ( \frac{A}{x - 1} + \frac{B}{x + 4} ). To find ( A ) and ( B ), multiply both sides by the common denominator ( (x - 1)(x + 4) ), then equate the numerators. After finding ( A ) and ( B ), integrate each term separately. The integral of ( \frac{A}{x - 1} ) will be ( A\ln|x - 1| ), and the integral of ( \frac{B}{x + 4} ) will be ( B\ln|x + 4| ), plus a constant ( C ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.