What is the integral of (4x^3+6)ln(x) ?

The integral of (4x^3+6)ln(x) is x^4ln(x)-(x^4)/4+6(xln(x)-x)+C

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integral of (4x^3+6)ln(x)

\int (4 x^{3} + 6) ln (x) d x

x^{4} ln (x) - \frac{x ^{4}}{4} + 6 (x ln (x) - x) + C

Solution steps

\int (4 x^{3} + 6) ln (x) d x

Expand

(4 x^{3} + 6) ln (x) : 4 x^{3} ln (x) + 6 ln (x)

= \int 4 x^{3} ln (x) + 6 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 4 x^{3} ln (x) d x + \int 6 ln (x) d x

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

\int 6 ln (x) d x = 6 (x ln (x) - x)

= x^{4} ln (x) - \frac{x ^{4}}{4} + 6 (x ln (x) - x)

Add a constant to the solution

= x^{4} ln (x) - \frac{x ^{4}}{4} + 6 (x ln (x) - x) + C

What is the integral of (4x^3+6)ln(x) ?
The integral of (4x^3+6)ln(x) is x^4ln(x)-(x^4)/4+6(xln(x)-x)+C