# Evaluate the following definite integrals: ∫23x(x+2)(x+3)⋅dx - Mathematics and Statistics

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Sum

Evaluate the following definite integrals: int_2^3 x/((x + 2)(x + 3)). dx

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#### Solution

Let I = int_2^3 x/((x + 2)(x + 3)) dx

Let x/((x + 2)(x + 3)) = "A"/(x + 2) + "B"/(x + 3)  ...(i)

∴ x = A(x + 3) + B(x + 2)        ...(ii)

Putting x = – 3 in (ii), we get,

∴ B = 3

Putting x = – 2 in (ii),we get,

∴  A = – 2

From (i), we get,

x/((x + 2)(x + 3)) = (–2)/(x + 2) + (3)/(x + 3)

∴ I = int_2^3 [(–2)/(x + 2) + 3/(x + 3)].dx

∴ I = –2int_2^3 (1)/(x + 2).dx + 3 int_2^3 (1)/(x + 3).dx

∴ I = –2[log|x + 2|]_2^3 + 3[log|x + 3|]_2^3

∴ I = –2log[log 5 – log 4] + 3[log 6 – log 5]

∴ I = –2[log(5/4)] + 3[log(6/5)]

∴ I = 3log(6/5) – 2log(5/4)

∴ I = log(6/5)^3 – 2log(5/4)^2

∴ I = log(216/125) – log(25/16)

∴ I = log(216/125 × 16/25)

∴ I = log(3456/3125).

Concept: Fundamental Theorem of Integral Calculus
Is there an error in this question or solution?
Chapter 6: Definite Integration - Exercise 6.1 [Page 145]

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