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Question
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Solution
Let I = `int (1)/(1 + x - x^2).dx`
∴ = `I = int1/(1 - (x^2 - x))dx`
∴ = `I = int1/(1-(x^2 - x + 1/4 - (1)/(4)))dx`
∴ = `I = int1/ ((1+1/4) - (x^2 - x + (1/2)^2))dx`
∴ = `I = int 1/ ((sqrt5/2)^2 - (x - 1/2)^2)dx` ...[`int(1/(a^2 - x^2dx) = 1/(2a) log |(a + x)/(a - x)|+c)`]
∴ `I = (1)/(2(sqrt(5)/2))log|(sqrt(5)/(2) + (x - 1/2))/(sqrt(5)/(2) - (x - 1/2))| + c`
∴ `I = (1)/sqrt(5) log |(sqrt(5) - 1 + 2x)/(sqrt(5) + 1 - 2x)|+ c`.
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