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in Definite Integrals by (65.2k points)

Evaluate: \(\int_2^3\frac{1}{(x + 1)(x + 2)}dx\)

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\(\int_2^3\frac{1}{(x + 1)(x + 2)}dx\)

Let 1 = A(x + 2) + B(x + 1) 

put x = -1 1 = A(-1 + 2) + B(0) ⇒ A = 1 

Put x = -2, 1 = A(0) + B(-2 + 1) → 1 = – B ⇒ B = -1

\(\int_2^3\frac{1.dx}{(x + 1)(x + 2)}\) = [1. log(3 + 1) – 1 log (3 + 2)] – [log(2 + 1) – log(2 + 2)] 

= log 4 – log 5 – log 3 + log 4 = 2 log 4 – log (5 × 3) = log 42 – log 15 = log (\(\frac{16}{15}\))

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