integral of x*ln^4(x)

Lösung

\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

Schritte zur Lösung

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

Wende die partielle Integration an

= \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})

Vereinfache

= \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}

Füge eine Konstante zur Lösung hinzu

= \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})

in v erse l a pl a ce \frac{16}{s ^{2} + 9}

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

\frac{d}{d x} (- \frac{1}{2} x^{4})

x \to 4 lim (3 x^{3})