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Question
Find `int (2x)/((x^2 + 1)(x^4 + 4))`dx
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Solution
Let x2 = 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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