Evaluate the integral. (Complete the square if necessary) `int (x-1)/(sqrt(x^2 - 2x)) dx`

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mathsworkmusic | (Level 2) Educator

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To solve this integral make a secant substitution:

`x-1 = sec theta`  so that `dx = sec theta tan theta d theta`

Rewrite the integral then as

`I = int (sec theta)/sqrt(sec^2 theta - 1) sec theta tan theta d theta`  = `int (sec^2 theta tan theta)/sqrt(sec^2 theta - 1) d theta `  `= int sec^2 theta d theta`  

since `x^2 - 2x = (x-1)^2 - 1` (completing the square in the denominator of the integrand as suggested), and since `sqrt(sec^2 theta - 1) = tan theta`

Now,

`int sec^2 theta d theta = tan theta + c`    where `c` is a constant of integration,  giving

`I = tan theta + c`

``Putting `x - 1 = sec theta`  back into this, note that  `(x-1)^2 -1 = sec^2 theta - 1 ` `= tan theta` , so that `tan theta = sqrt((x-1)^2 - 1) = sqrt(x^2 - 2x)`

This gives then that

`I = sqrt(x^2 - 2x) + c`    where `c` is a constant of integration

NB see Example 6 and solution in the reference link below

Sources:

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