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

The integral of x*ln^3(x) 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^3(x)

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

x \to 2 lim (2 x^{2} - 3 x + 5)

\frac{s ^{2}}{( s ^{2} + \frac{1}{4} )}

\int \frac{( x + 1 ) ^{2}}{x} d x

sin (3 x)

y = e^{x} + 1

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