What is the integral of xln(x+6) ?

The integral of xln(x+6) is ln(x+6)(1/2 (x+6)^2-6(x+6))-1/4 (x^2-12x-108)+C

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integral of xln(x+6)

\int x ln (x + 6) d x

ln (x + 6) (\frac{1}{2} (x + 6)^{2} - 6 (x + 6)) - \frac{1}{4} (x^{2} - 12 x - 108) + C

Solution steps

\int x ln (x + 6) d x

Apply u-substitution

: \int (u - 6) ln (u) d u

= \int (u - 6) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} - 6 u) - \int \frac{1}{2} (u - 12) d u

= ln (u) (\frac{u ^{2}}{2} - 6 u) - \int \frac{1}{2} (u - 12) d u

\int \frac{1}{2} (u - 12) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 12 u)

= ln (u) (\frac{u ^{2}}{2} - 6 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 12 u)

Substitute back

u = x + 6

= ln (x + 6) (\frac{( x + 6 ) ^{2}}{2} - 6 (x + 6)) - \frac{1}{2} (\frac{( x + 6 ) ^{2}}{2} - 12 (x + 6))

Simplify

ln (x + 6) (\frac{( x + 6 ) ^{2}}{2} - 6 (x + 6)) - \frac{1}{2} (\frac{( x + 6 ) ^{2}}{2} - 12 (x + 6)) : ln (x + 6) (\frac{1}{2} (x + 6)^{2} - 6 (x + 6)) - \frac{1}{4} (x^{2} - 12 x - 108)

= ln (x + 6) (\frac{1}{2} (x + 6)^{2} - 6 (x + 6)) - \frac{1}{4} (x^{2} - 12 x - 108)

Add a constant to the solution

= ln (x + 6) (\frac{1}{2} (x + 6)^{2} - 6 (x + 6)) - \frac{1}{4} (x^{2} - 12 x - 108) + C

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What is the integral of xln(x+6) ?
The integral of xln(x+6) is ln(x+6)(1/2 (x+6)^2-6(x+6))-1/4 (x^2-12x-108)+C