What is the integral of x*ln(3x) ?

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

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integral of x*ln(3x)

\int x \cdot ln (3 x) d x

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

Solution steps

\int x ln (3 x) d x

Apply Integration By Parts

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

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

Add a constant to the solution

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

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