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Question
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
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Solution
Let `I = int (2x)/((x^2 + 1)(x^2 + 3))` dx
Putting x2 = t, 2x dx = dt
`therefore I = int dt/((t + 1)(t + 3))`
Now, `1/((t + 1)(t + 3)) = A/(t + 1) = B/(t + 3)`
1 = A(t + 3) + B(t + 1)
Put t = -1
1 = A(-1 + 3)
⇒ 1 = 2A
∴ A `= 1/2`
Put t = -3
1 = B (-3 + 1)
⇒ 1 = -2B
∴ B `= -1/2`
`therefore 1/((t + 1)(t + 3)) = 1/(2(t + 1)) - 1/(2(t + 3))`
`therefore I = int 1/((t + 1)(t + 3)) dt = 1/2 int 1/(t + 1) dt - 1/2 int 1/(t + 3) dt`
`= 1/2 log (t + 1) - 1/2 log (t + 3) + C`
`= 1/2 log abs ((t + 1)/(t + 3)) + C`
`= 1/2 log abs ((x^2 + 1)/(x^2 + 3)) + C`
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