Answer 1

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
– 2 = A
∴ 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)]*dx`

= `-2int_2^3 (1)/(x + 2)*dx + 3 int_2^3 (1)/(x + 3)*dx`

= `-2[log|x + 2|]_2^3 + 3[log|x + 3|]_2^3`

= `-2log[log 5 – log 4] + 3[log 6 – log 5]`

= `-2[log(5/4)] + 3[log(6/5)]`

= `3log(6/5) - 2log(5/4)`

= `log(6/5)^2 - 2log(5/4)^2`

= `log(216/125) - log(25/16)`

= `log(216/125 xx 16/25)`

∴ I = `log(3456/3125)`.