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

The integral of x*ln^4(x) 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^4(x)

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

x \to 2 - lim (- \frac{1}{x - 2})

\frac{d}{d x} (- 4 cos (\frac{x}{2}))

y^{^{''}} - y^{^{'}} - 30 y = 0

\frac{16}{s ^{2} + 9}

\frac{\partial}{\partial y} (\frac{4 x}{y})

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