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Question
Integrate the function `(x + 2)/sqrt(x^2 -1)`
Solution
Let `I = (x + 2)/sqrt(x^2 - 1) dx`
`= int x/sqrt(x^2 - 1) dx + int 2/sqrt(x^2 - 1) dx`
`= I_1 + I_2 + C` (Say) ....(i)
Now `I_1 = int x/sqrt(x^2 - 1) dx`
Put x2 - 1 = t
2x dx = dt
`= 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` ....(ii)
and `I_2 = int 2/sqrt(x^2 - 1) dx`
`= 2 log abs (x + sqrt(x^2 - 1) + C_2` .....(iii)
From (i), (ii) and (iii), we get,
`therefore I = sqrt(x^2 - 1) + 2 log abs (x + sqrt(x^2 - 1)) + C`
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