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

The integral of (x^2+3x+2)ln(x) is ln(x)((x^3)/3+(3x^2)/2+2x)-1/6 ((2x^3)/3+(9x^2)/2+12x)+C

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

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

ln (x) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 2 x) - \frac{1}{6} (\frac{2 x ^{3}}{3} + \frac{9 x ^{2}}{2} + 12 x) + C

Solution steps

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

Apply Integration By Parts

: ln (x) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 2 x) - \int \frac{1}{6} (2 x^{2} + 9 x + 12) d x

= ln (x) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 2 x) - \int \frac{1}{6} (2 x^{2} + 9 x + 12) d x

\int \frac{1}{6} (2 x^{2} + 9 x + 12) d x = \frac{1}{6} (\frac{2 x ^{3}}{3} + \frac{9 x ^{2}}{2} + 12 x)

= ln (x) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 2 x) - \frac{1}{6} (\frac{2 x ^{3}}{3} + \frac{9 x ^{2}}{2} + 12 x)

Add a constant to the solution

= ln (x) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 2 x) - \frac{1}{6} (\frac{2 x ^{3}}{3} + \frac{9 x ^{2}}{2} + 12 x) + C

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

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\int \frac{x ^{3}}{x - 4} d x

What is the integral of (x^2+3x+2)ln(x) ?
The integral of (x^2+3x+2)ln(x) is ln(x)((x^3)/3+(3x^2)/2+2x)-1/6 ((2x^3)/3+(9x^2)/2+12x)+C