Question

Integrate the following with respect to x.

`(3x^2 - 2x + 5)/((x - 1)(x^2 + 5))`

Answer 1

`(3x^2 - 2x + 5)/((x - 1)(x^2 + 5)) = "A"/(x - 1) + ("B"x + "C")/(x^2 + 5)`

⇒ 3x² − 2x + 5 = A(x² + 5) + (Bx + C)(x − 1)  .......(1)

Put x  1 on both sides of (1)

3 − 2 + 5 = A(1 + 5)

6 = 6A

⇒ A = 1

Put x = 0 on both sides of (1)

5 = 5 − C

C = 0

Put x = 2 on both sides of (1)

12 − 4 + 5 = 1(4 + 5) + (2B)(2 − 1)

13 = 9 + 2B

2B = 4

B = 2

So `(3x^2 - 2x + 5)/((x - 1)(x^2 + 5))  "d"x = int 1/(x - 1)  "d"x + int (2x)/(x^2 + 5)  "d"x + "c"`

Now `int (3x^2 - 2x + 5)/((x - 1)(x^2 + 5))  "d"x = int 1/(x - 1)  "d"x + (2x)/(x^2 + 5)  "d"x + "c"`

Now `int (2x)/(x^2 + 5) = int ("d"(x^2 + 5))/(x^2 + 5)`

= `log|x^2 + 5|`

⇒ The solution is = `log|x - 1| + log|x^2 + 5| + "c"`

= `log|(x - 1)(x^2 + 5)| + "c"`

= `log|x^3 - x^2 + 5x - 5| + "c"`