Question

Evaluate `int_2^3 x/((x + 2)(x + 3))  "d"x`

Answer 1

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)`