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

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

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integral of xln(3x^3)

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{3 x}{2} d x = \frac{3 x ^{2}}{4}

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

Simplify

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

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

Add a constant to the solution

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

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