Question

Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`

Answer 1

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

Let `1/((x^2 + 1)(x^2 + 2)) = A/(x^2 + 1) + B/(x^2 + 2)`

⇒ 1 = A(x² + 2) + B(x² + 1)

⇒ 1 = (A + B)x² + (2A + B)

On comparing both sides, we get

A + B = 0 and 2A + B = 0

On solving the above equations, we get

A = 1 and B = –1

∴ I = `int(1/(x^2 + 1) - 1/(x^2 + 2))2xdx`

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

I = `log|x^2 + 1| - log|x^2 + 2| + C`

I = `log|(x^2 + 1)/(x^2 + 2)| + C`