What is the integral of (x^2+2x)ln(3x) ?

The integral of (x^2+2x)ln(3x) is 1/3 x^3ln(3x)-(x^3)/9+x^2ln(3x)-(x^2)/2+C

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integral of (x^2+2x)ln(3x)

\int (x^{2} + 2 x) ln (3 x) d x

\frac{1}{3} x^{3} ln (3 x) - \frac{x ^{3}}{9} + x^{2} ln (3 x) - \frac{x ^{2}}{2} + C

Solution steps

\int (x^{2} + 2 x) ln (3 x) d x

Expand

(x^{2} + 2 x) ln (3 x) : x^{2} ln (3 x) + 2 x ln (3 x)

= \int x^{2} ln (3 x) + 2 x ln (3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} ln (3 x) d x + \int 2 x ln (3 x) d x

\int x^{2} ln (3 x) d x = \frac{1}{3} x^{3} ln (3 x) - \frac{x ^{3}}{9}

\int 2 x ln (3 x) d x = x^{2} ln (3 x) - \frac{x ^{2}}{2}

= \frac{1}{3} x^{3} ln (3 x) - \frac{x ^{3}}{9} + x^{2} ln (3 x) - \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{1}{3} x^{3} ln (3 x) - \frac{x ^{3}}{9} + x^{2} ln (3 x) - \frac{x ^{2}}{2} + C

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What is the integral of (x^2+2x)ln(3x) ?
The integral of (x^2+2x)ln(3x) is 1/3 x^3ln(3x)-(x^3)/9+x^2ln(3x)-(x^2)/2+C