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

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

integral of 4x^3*ln(x)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

= 4 (\frac{1}{4} x^{4} ln (x) - \int \frac{x ^{3}}{4} d x)

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= 4 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})

Simplify

4 (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) : x^{4} ln (x) - \frac{x ^{4}}{4}

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

Add a constant to the solution

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

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