Question

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

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,

∴ 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)`.