Question

Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`

Answer 1

Let I = `int (x^2 + 3)/((x^2 - 1)(x^2 - 2)).dx`

`(x^2 + 3)/((x^2 - 1)(x^2 - 2)) = "A"/(x^2 - 1) + "B"/(x^2 - 2)`

∴ x² + 3 = A(x² - 2) + B(x² - 1)

Put x² - 1 = 0 i. e. x² = 1

∴ 1 + 3 = A(1 - 2) + B(1 - 1)

∴ 1 + 3 = A(1 - 2) + 0

∴ 4 = A × -1

∴ A = - 4

Put x² - 2 = 0 i. e. x² = 2

∴ `2 + 3 = 0 + B (2 - 1)`

∴ 5 = B × 1

∴ B = 5

I = `int (- 4)/(x^2 - 1^2) "dx" + int 5/(x^2 - (sqrt(2))^2)` dx

I = `- 4 xx 1/(2 xx 1) log |(x - 1)/(x + 1)| + 5 xx 1/(2 xx sqrt2)  log  |(x - sqrt2)/(x + sqrt2)|` + c    ...`[int 1/(x^2 - a^2) dx = 1/(2a) log |(x - a)/(x + a)| + "c"]`

I = `- 2 log |(x - 1)/(x + 1)| + 5/(2 sqrt2) log |(x - sqrt2)/(x + sqrt2)| + c`