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

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

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integral of x(ln(x))^4

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

\frac{1}{2} x^{2} ln^{4} (x) - x^{2} ln^{3} (x) + \frac{3}{2} x^{2} ln^{2} (x) - \frac{3}{2} x^{2} ln (x) + \frac{3}{4} x^{2} + C

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

= \frac{1}{2} x^{2} ln^{4} (x) - x^{2} ln^{3} (x) + \frac{3}{2} x^{2} ln^{2} (x) - \frac{3}{2} x^{2} ln (x) + \frac{3}{4} x^{2} + C

\int \frac{2}{( x + 1 ) ( x + 2 ) ( x + 3 )} d x

\int x (3 x^{2} + 5)^{9} d x

\int \frac{2}{v v ^{2} - 16} d v

\int \frac{2 x}{3 x ^{2} + 1} d x

\frac{1}{u ^{2}}

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