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Question
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Solution
Let I = `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Let `1/((x^2 + 1)(x^2 + 2)) = A/(x^2 + 1) + B/(x^2 + 2)`
⇒ 1 = A(x2 + 2) + B(x2 + 1)
⇒ 1 = (A + B)x2 + (2A + B)
On comparing both sides, we get
A + B = 0 and 2A + B = 0
On solving the above equations, we get
A = 1 and B = –1
∴ I = `int(1/(x^2 + 1) - 1/(x^2 + 2))2xdx`
I = `int (2x)/(x^2 + 1) dx - int (2x)/(x^2 + 2) dx`
I = `log|x^2 + 1| - log|x^2 + 2| + C`
I = `log|(x^2 + 1)/(x^2 + 2)| + C`
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