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

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

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integral of (x+6)ln(x)

\int (x + 6) ln (x) d x

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

Solution steps

\int (x + 6) ln (x) d x

Apply Integration By Parts

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

\int \frac{1}{2} (x + 12) d x = \frac{1}{2} (\frac{x ^{2}}{2} + 12 x)

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

Add a constant to the solution

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

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