Question

Evaluate:

`int (2x + 7)/(x^2 - x - 2) dx`

Answer 1

`int (2x + 7)/(x^2 - x - 2) dx`

= `int (2x + 7)/(x^2 - 2x + x - 2) dx`

= `int (2x + 7)/((x - 2)(x + 1)) dx`

Let `(2x + 7)/((x - 2)(x + 1)) = A/(x - 2) + B/(x + 1)`

⇒ `(2x + 7)/((x - 2)(x + 1)) = (A(x + 1) + B(x - 2))/((x - 2)(x + 1))`

⇒ 2x + 7 = (A + B)x + A – 2B

Comparing both sides, we get

A + B = 2 and A – 2B = 7

Solving these equations for A and B, we get

`A = 11/3` and `B = -5/3`

Now, `int (2x + 7)/((x - 2)(x + 1)) dx = 11/3 int 1/((x - 2)) dx - 5/3 int 1/(x + 1) dx`

= `11/3 log |x - 2| - 5/3 log |x + 1| + c`