integral of (ln(ln(x))ln(x))/x

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int u ln (u) d u

Wende die partielle Integration an

= \frac{1}{2} u^{2} ln (u) - \int \frac{u}{2} d u

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

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

Setze in

u = ln (x)

ein

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{5 x ^{2} + 7 x + 6}{x ^{2} + x + 1} d x

\int \frac{x ^{2} + 7 x + 2}{( x + 1 ) ( x - 2 ) ( x + 3 )} d x

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

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

\int (x x - 5) d x