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Question
Integrate the following with respect to x.
`1/sqrt(9x^2 - 7)`
Solution
`int (1 "d"x)/sqrt(9x^2 - 7) = int ("d"x)/sqrt(9(x^2 - 7/9)`
= `1/3 int ("d"x)/sqrt(x^2 - 7/9)`
= `1/3 int ("d"x)/sqrt(x^2 - (sqrt(7)/3)^2`
= `1/3 log |x + sqrt(x^2 - (sqrt(7)/3)^2)| + "c"`
= `1/3 log|x + sqrt((9x^2 - 7)/9)| + "c"`
= `1/3 log |3x + sqrt(9x^2 - 7)| + "c" - 1/3 log 9`
= `1/3 log |3x + sqrt(9x^2 - 7)| + "k"`
Where k = `"C" - 1/3 log 9`
Which is a constant
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