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Question
Integrate the function `(x - 1)/sqrt(x^2 - 1)`
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Solution
Let `I = int (x - 1)/sqrt(x^2 - 1) dx`
`= int x/(sqrt(x^2 - 1)) dx - int 1/sqrt(x^2 - 1) dx`
`= I_1 - I_2` (say)
Now , `I_1 = x/sqrt(x^2 - 1) dx`
Put x2 - 1 = t
2x dx = dt ⇒ x dx = `1/2` dt
`therefore I = 1/2 int dt/sqrtt = 1/2 int t^((-1)/2) dt`
`= 1/2 xx t^(1/2)/(1/2) + C_1 = sqrtt = C_1`
`= sqrt(x^2 - 1) + C_1`
and `I_2 int 1/sqrt(x^2 - 1) dx`
`= log [x + sqrt(x^2) - 1] + C_2` `....[∵ int dx/sqrt(x^2 - a^2) = log |x + sqrt(x^2 - a^2)| + C]`
`therefore I = sqrt(x^2 - 1) - log |x + sqrt(x^2 - 1)| +C`
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