Question

Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`

Answer 1

Let I = `int (2x)/((2 + x^2)(3 + x^2))*dx`
Put x² = t
∴ 2x dx = dt

∴ I = `int (1)/((2 + t)(3 + t))*dt`

= `int ((3 + t) - (2 + t))/((2 + t)(3 + t))*dt`

= `int [1/(2 + t) - 1/(3 + t)]*dt`

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

= log|2 + t| – log|3 + t| + c

= `log|(2 + t)/(3 + t)| + c`

= `log|(2 + x^2)/(3 + x^2)| + c`.