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

The integral of-x^3ln(x) is -1/4 x^4ln(x)+(x^4)/(16)+C

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integral of-x^3ln(x)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

= - (\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}

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

Simplify

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

Add a constant to the solution

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

What is the integral of-x^3ln(x) ?
The integral of-x^3ln(x) is -1/4 x^4ln(x)+(x^4)/(16)+C