What is the integral of xln(x+3) ?

The integral of xln(x+3) is ln(x+3)(1/2 (x+3)^2-3(x+3))-1/4 (x^2-6x-27)+C

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integral of xln(x+3)

\int x ln (x + 3) d x

ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27) + C

Solution steps

\int x ln (x + 3) d x

Apply u-substitution

: \int (u - 3) ln (u) d u

= \int (u - 3) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} - 3 u) - \int \frac{1}{2} (u - 6) d u

= ln (u) (\frac{u ^{2}}{2} - 3 u) - \int \frac{1}{2} (u - 6) d u

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

= ln (u) (\frac{u ^{2}}{2} - 3 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 6 u)

Substitute back

u = x + 3

= ln (x + 3) (\frac{( x + 3 ) ^{2}}{2} - 3 (x + 3)) - \frac{1}{2} (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3))

Simplify

ln (x + 3) (\frac{( x + 3 ) ^{2}}{2} - 3 (x + 3)) - \frac{1}{2} (\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3)) : ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27)

= ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27)

Add a constant to the solution

= ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27) + C

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What is the integral of xln(x+3) ?
The integral of xln(x+3) is ln(x+3)(1/2 (x+3)^2-3(x+3))-1/4 (x^2-6x-27)+C