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

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#### Solution

Let x^{2} = t

2xdx = dt

`=> dx = dt/(2x)`

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

`= int (dt)/((t +1)(t^2 + 4))`

Let `1/((t + 1)(t^2 + 4)) = A/(t + 1) + (Bt + C)/(t^2+4)` .....(1)

`=> 1 = A(t^2 + 4) + (Bt + C) (t + 1)` ...(2)

Putting t=−1 in (2)

1 = A(1 + 4) +0

⇒5A = 1

`=> A = 1/5`

Putting t = 0 in 2

4A + C = 1

`=>C = 1 - 4/5`

`=> C = 1/5`

Putting t = 1 in (2)

1 = 5A + 2B + 2C

`=> -2B = 2/5`

`=>B = (-1)/5`

Putting the values of A, B and C in (1)

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