What is the integral of ln(1/(x-2)) ?

The integral of ln(1/(x-2)) is -xln(x-2)+2ln(x-2)+x-2+C

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integral of ln(1/(x-2))

\int ln (\frac{1}{x - 2}) d x

- x ln (x - 2) + 2 ln (x - 2) + x - 2 + C

Solution steps

\int ln (\frac{1}{x - 2}) d x

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= \int - ln (x - 2) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int ln (x - 2) d x

Apply u-substitution

: \int ln (u) d u

= - \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= - (u ln (u) - \int 1 d u)

\int 1 d u = u

= - (u ln (u) - u)

Substitute back

u = x - 2

= - ((x - 2) ln (x - 2) - (x - 2))

Simplify

- ((x - 2) ln (x - 2) - (x - 2)) : - x ln (x - 2) + 2 ln (x - 2) + x - 2

= - x ln (x - 2) + 2 ln (x - 2) + x - 2

Add a constant to the solution

= - x ln (x - 2) + 2 ln (x - 2) + x - 2 + C

\int 1 0^{- y} d y

\frac{d}{d x} (2 x^{3} - 15 x^{2} + 36 x - 10)

\int (t - \frac{1}{t}) d t

\frac{d}{d x} (sin (7 x + ln (5 x)))

x \to 0 lim (\frac{1}{x ^{2} ( x + 7 )})

What is the integral of ln(1/(x-2)) ?
The integral of ln(1/(x-2)) is -xln(x-2)+2ln(x-2)+x-2+C