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Question
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Solution
Let I = `int (1)/(5 - 4x - 3x^2).dx`
5 – 4x –3x2 = `[5/3 - (x^2 + 4/3 x)]`
= `3[(5)/(3) - (x^2 + (4)/(3)x + (4)/(9)) + 4/9]`
= `3[(19)/(9) - (x^2 + (4x)/(3) + (4)/(9))]`
= `3[(sqrt(19)/3)^2 - (x + 2/3)^2]`
I = `int (1)/(3[(sqrt(19)/3)^2 - (x + 2/3)^2]).dx`
= `(1)/(3) (1)/(2(sqrt(19)/3))log |(sqrt(19)/(3) + (x + 2/3))/(sqrt(19)/(3) - (x + 2/3))| + c`
= `(1)/(2sqrt(19))log |(sqrt(19) + 2 + 3x)/(sqrt(19) - 2 - 3x)| + c`
= `(1)/(2sqrt(19))log |(3x + 2 + sqrt(19))/(-(3x + 2 - sqrt(19)))| + c`
= `(1)/(2sqrt(19))log |(3x + 2 + sqrt(19))/(3x + 2 - sqrt(19))| + c`. ...[∵ | – x |= x]
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