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Evaluate the following : ∫15-4x-3x2.dx - Mathematics and Statistics

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Question

Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`

Sum
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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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Chapter 3: Indefinite Integration - Exercise 3.2 (B) [Page 123]

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