Question

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

Answer 1

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

Using partial fraction

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

2x + 1 = A (x - 2) + B( x + 1)

Put x = 2 in equation (i),

4 + 1 = B ( 2 + 1)

B = `5/3`

Put x = -1 in equation (i)

-2 + 1 = A[-1 -2]

A = `(-1)/(-3)`

A = `1/3`

Substituting the values of A and B in equation (α) we get

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

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

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

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