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

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

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integral of xln(3x)-x

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

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

Solution steps

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

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

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

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

Add a constant to the solution

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

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