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Question
Evaluate `int_2^3 x/((x + 2)(x + 3)) "d"x`
Solution
Let I = `int_2^3 x/((x + 2)(x + 3)) "d"x`
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
– 3 = – B
∴ B = 3
Putting x = – 2 in (ii), we get
– 2 = A
∴ 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)] "d"x`
= `-2 int_2^3 1/(x + 2) "d"x + 3 int_2^3 1/(x + 3) "d"x`
= `-2[log|x + 2|]_2^3 + 3[log|x + 3|]_2^3`
= – 2(log 5 – log 4) + 3(log 6 – log 5)
= `- 2 log(5/4) + 3 log(6/5)`
= `3 log(6/5) - 2log(5/4)`
= `log (6/5)^3 - log(5/4)^2`
= `log(216/125) - log(25/16)`
= `log(216/125 xx 16/25)`
∴ I = `log(3456/3125)`
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