Question

Find: `int(2x)/((x^2+1)(x^2-4)) dx`

Answer 1

Let `I = int(2x)/((x^2+1)(x^2-4)) dx`

Put x² = y ⇒ 2x.dx = dy

`I = int dy/ ((y+1)(y-4))`

`int (A/(y+1)+B/(y-4)).dy`

Let `A/(y+1)+B/(y-4)=1/((y+1)(y-4))`

⇒ A(y − 4) + B(y + 1) = 1

⇒ (A + B)y − 4A + B = 1 + 0y

After equating, we get

A + B = 0 & B − 4A = 1

Solving we get,

`A=(-1)/5 &  B = 1/5`

put value of A & B in I, we get

`I = (-1)/5 int dy/(y+1) + 1/5 int dy/(y-4)`

`= (-1)/5 log (y+1) + 1/5 log (y-4) + C`

`= (-1)/5 log (x^2+1)+1/5 log(x^2-4) + C`