Question

Integrate the rational function:

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

Answer 1

Let `I = int (2x)/((x^2 + 1)(x^2 + 3))` dx

Putting x² = t, 2x dx = dt

`therefore I = int dt/((t + 1)(t + 3))`

Now, `1/((t + 1)(t + 3)) = A/(t + 1) = B/(t + 3)`

1 = A(t + 3) + B(t + 1)

Put t = -1

1 = A(-1 + 3)

⇒ 1 = 2A  

∴ A `= 1/2`

Put t = -3

1 = B (-3 + 1)

⇒ 1 = -2B   

∴ B `= -1/2`

`therefore 1/((t + 1)(t + 3)) = 1/(2(t + 1)) - 1/(2(t + 3))`

`therefore I = int 1/((t + 1)(t + 3))  dt = 1/2 int 1/(t + 1)   dt - 1/2 int 1/(t + 3)  dt` 

`= 1/2  log (t + 1) - 1/2  log (t + 3) + C`

`= 1/2  log abs ((t + 1)/(t + 3)) + C`

`= 1/2  log abs ((x^2 + 1)/(x^2 + 3)) + C`