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

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mathsworkmusic | (Level 2) Educator

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To evaluate this integral, first complete the square (under the square-root sign) in the denominator of the integrand, as suggested, giving:

`I = int 1/((x-1)sqrt((x-1)^2 - 1)) dx`

Notice now that the integrand has the element `1/sqrt(y^2 - a^2)`   in it, which suggests that a secant substitution would be sensible.

So use the substitution  `x-1 = sec theta`  so that  `dx = sec theta tan theta d theta`

Now rewrite the integral as

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

Knowing the trigonometric relation  `tan theta = sqrt(sec^2 theta - 1)` ,  we then have that

`I = int (sec theta tan theta)/ (sec theta tan theta) d theta = int 1. d theta`  `= theta` + c

Putting `x - 1 = sec theta`  back in, establish `theta` in terms of `x`

`theta = sec^(-1)(x-1) = arcsec (x-1)`   where arcsec is the inverse secant function.

Therefore `I = arcsec(x-1) + c`  where `c` is a constant of integration.

NB see Example 6 in the first reference link below

Sources:
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gsenviro | College Teacher | (Level 1) Educator Emeritus

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You may want to try the online wolfram calculator: http://integrals.wolfram.com/index.jsp?expr=1%2F%28%28x-1%29Sqrt%5Bx%5E2-2x%5D%29&random=false

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