Question

Integrate the rational function:

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

Answer 1

`2/((1 - x)(1 + x^2)) = A/(1 - x) = (Bx + C)/(1 + x^2)`

2 = A(1 + x²) + (1 - x) Bx + C

Put x = 1

2 = 2A + 0

⇒ A = 1

Put x = 0

2 = A + C

⇒ C = 1

Comparing the coefficients of x² on both sides,

0 = A - B

⇒ B = A = 1

`therefore 2/((1 - x)(1 + x^2)) = 1/(1 - x) + (x + 1)/(1 + x^2)`

`= 1/(1 - x) + x/(1 + x^2) + 1/(1 + x^2)`

On integrating

`int 2/((1 - x)(1 + x^2))  dx`

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

`= - log abs (1 - x) + 1/2  log abs (1 + x^2) + tan^-1 x + C`