integral of ((ln(x))/x)^3

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e 4 x^{2} + 25

\frac{d x}{d t} = - 0.1 (x - 100 + 40 e^{- 0.1 t})

x \to 2 - lim (ln ∣ x - 2 ∣)

\int \frac{1}{x ^{2} - y ^{2}} d y

d er i v a t i v e f (x) = \frac{x ^{2} + 2 x - 6}{( x + 1 ) ^{2}}