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

The integral of x(ln(x))^3 is 1/2 x^2ln^3(x)-3/8 (2x^2ln^2(x)-2x^2ln(x)+x^2)+C

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integral of x(ln(x))^3

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

\frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2}) + C

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

\int \frac{3}{2} x ln^{2} (x) d x = \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})

= \frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})

Add a constant to the solution

= \frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2}) + C

\int \frac{e ^{3 x}}{1 + e ^{x}} d x

x \to 3 lim (x^{2} + 3 x - 1)

\int e^{2} cos (3 x) d x

\frac{d}{d x} (cos (arcsin (3 x)))

\frac{d x}{d t} = 6 - \frac{7 x}{600 + 2 t}

What is the integral of x(ln(x))^3 ?
The integral of x(ln(x))^3 is 1/2 x^2ln^3(x)-3/8 (2x^2ln^2(x)-2x^2ln(x)+x^2)+C