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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

x \to 0 lim (3 (x^{\frac{x}{2}}))

f (x) = \frac{6}{5 x + 3}, at x = - 1

x^{^{'}} = x + 1

f (x) = 9 x e^{- k x}

- x^{- 3}

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^2ln(x)+3xln(x)-(x^3)/9+(x^2)/2-3x+C