What is the integral of (2x+6)ln(x) ?

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

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integral of (2x+6)ln(x)

\int (2 x + 6) ln (x) d x

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

Solution steps

\int (2 x + 6) ln (x) d x

Apply Integration By Parts

: ln (x) (x^{2} + 6 x) - \int x + 6 d x

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

\int x + 6 d x = \frac{x ^{2}}{2} + 6 x

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

Simplify

ln (x) (x^{2} + 6 x) - (\frac{x ^{2}}{2} + 6 x) : x^{2} ln (x) + 6 x ln (x) - \frac{x ^{2}}{2} - 6 x

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

Add a constant to the solution

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

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