What is the integral of xln(x-2) ?

The integral of xln(x-2) is ln(x-2)(1/2 (x-2)^2+2(x-2))-1/4 (x^2+4x-12)+C

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integral of xln(x-2)

\int x ln (x - 2) d x

ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12) + C

Solution steps

\int x ln (x - 2) d x

Apply u-substitution

: \int (u + 2) ln (u) d u

= \int (u + 2) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} + 2 u) - \int \frac{1}{2} (u + 4) d u

= ln (u) (\frac{u ^{2}}{2} + 2 u) - \int \frac{1}{2} (u + 4) d u

\int \frac{1}{2} (u + 4) d u = \frac{1}{2} (\frac{u ^{2}}{2} + 4 u)

= ln (u) (\frac{u ^{2}}{2} + 2 u) - \frac{1}{2} (\frac{u ^{2}}{2} + 4 u)

Substitute back

u = x - 2

= ln (x - 2) (\frac{( x - 2 ) ^{2}}{2} + 2 (x - 2)) - \frac{1}{2} (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2))

Simplify

ln (x - 2) (\frac{( x - 2 ) ^{2}}{2} + 2 (x - 2)) - \frac{1}{2} (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2)) : ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12)

= ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12)

Add a constant to the solution

= ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12) + C

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What is the integral of xln(x-2) ?
The integral of xln(x-2) is ln(x-2)(1/2 (x-2)^2+2(x-2))-1/4 (x^2+4x-12)+C