What is the integral of (3x-1)ln(x) ?

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

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integral of (3x-1)ln(x)

\int (3 x - 1) ln (x) d x

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

Solution steps

\int (3 x - 1) ln (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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