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

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

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

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

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

Solution steps

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

Expand

(x^{2} + 2 x + 3) ln (x) : x^{2} ln (x) + 2 x ln (x) + 3 ln (x)

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

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

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

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

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

Add a constant to the solution

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

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