integral of (ln^2(x))/(x^3)

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

t an g e n t y = sin (2 x), at x = \frac{3 π}{4}

\int_{0}^{π} sin^{4} (x) d x

x^{2} y^{3} d x + x^{3} y^{2} d y = 0

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

x \to \infty lim (2 x^{3})