What is the integral of (3x^2+2x+4)ln(x) ?

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

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integral of (3x^2+2x+4)ln(x)

\int (3 x^{2} + 2 x + 4) ln (x) d x

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

Solution steps

\int (3 x^{2} + 2 x + 4) ln (x) d x

Expand

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

= \int 3 x^{2} ln (x) + 2 x ln (x) + 4 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 3 x^{2} ln (x) d x + \int 2 x ln (x) d x + \int 4 ln (x) d x

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

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

\int 4 ln (x) d x = 4 (x ln (x) - x)

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

Add a constant to the solution

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

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