Question

Evaluate : `int (2"x" + 1)/(("x" + 1)("x"-2))  "dx"` 

Answer 1

Let I = `int (2"x" + 1)/(("x" + 1)("x"-2))  "dx"` 

By partial fraction 

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

∴ 2x + 1= A (x - 2) + B(x + 1) 

Put x = -1

2 (-1) + 1 =  A(x - 2) + B (0) 

⇒ -1 = -3 A

∴ A = `1/3`

Put  x = 2

2(2) + 1 = A(O) + B(2+1) 

⇒ 5 = 3 B

∴ B = `5/3`

∴ I = `int [(1/3)/("x" + 1) + (5/3)/("x" - 2)] "dx"`

`= 1/3 int 1/("x" + 1) "dx" + 5/3 int 1/("x" - 2) "dx"`

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