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Question
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
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Solution
Let `I = int_ (2x+1)/((x^2+1)(x^2+4))dx`
Let `(2x+1)/((x^2+1)(x^2+4)) = (Ax + B)/(x^2 + 1) + (Cx + D)/(x^2 + 4)`
Getting A = `2/3, B = 1/3, C = (-2)/3, D = (-1)/3`
∴ `I = 2/3 int x/(x^2 + 1) dx + 1/3 int x/(x^2 + 1)dx + (- 2)/3 int (xdx)/(x^2 + 4) + (-1)/3 int dx/(x^2 + 4)`
= `1/3 log | x^2 + 1| + 1/3 tan^-1 x - 1/3 log | x^2 + 4| - 1/6 tan^-1 x/2 + C`.
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