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

The integral of (ln(x))^2x^3 is 1/4 x^4ln^2(x)-1/2 (1/4 x^4ln(x)-(x^4)/(16))+C

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

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

\frac{1}{4} x^{4} ln^{2} (x) - \frac{1}{2} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + C

Solution steps

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

Simplify

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

Apply Integration By Parts

: \frac{1}{4} x^{4} ln^{2} (x) - \int \frac{1}{2} x^{3} ln (x) d x

= \frac{1}{4} x^{4} ln^{2} (x) - \int \frac{1}{2} x^{3} ln (x) d x

\int \frac{1}{2} x^{3} ln (x) d x = \frac{1}{2} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})

= \frac{1}{4} x^{4} ln^{2} (x) - \frac{1}{2} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16})

Add a constant to the solution

= \frac{1}{4} x^{4} ln^{2} (x) - \frac{1}{2} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{16}) + C

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What is the integral of (ln(x))^2x^3 ?
The integral of (ln(x))^2x^3 is 1/4 x^4ln^2(x)-1/2 (1/4 x^4ln(x)-(x^4)/(16))+C