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Question
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
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Solution
`int (1)/sqrt(3x^2 + 8).dx`
= `(1)/sqrt(3) int (1)/sqrt(x^2 + 8/3).dx`
= `(1)/sqrt(3) int (1)/sqrt(x^2 + (sqrt(8/3))^2).dx`
= `(1)/sqrt(3) log |x + sqrt(x^2 + (sqrt(8/3))^2)| + c_1`
= `(1)/sqrt(3) log |x + sqrt(x^2 + 8/3)| + c_1`
= `(1)/sqrt(3) log |(sqrt(3)x + sqrt(3x^2 + 8))/sqrt(3)| + c_1`
= `(1)/sqrt(3) log |sqrt(3)x + sqrt(3x^2 + 8)| - logsqrt(3) + c_1`
= `(1)/sqrt(3) log |sqrt(3)x + sqrt(3x^2 + 8)| + c, "where" c = c_1 - logsqrt(3)`
Alternative Method :
`int (1)/sqrt(3x^2 + 8).dx`
= `int (1)/sqrt((sqrt(3)x)^2 + (sqrt(8))^2).dx`
= `(log|sqrt(3)x + sqrt((sqrt(3)x)^2 + sqrt((8))^2| + c))/sqrt(3)`
= `(1)/sqrt(3) log |sqrt(3)x + sqrt(3x^2 + 8)| + c`.
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