integral of (ln(x))/((1+ln(x))^2)

Lösung

\int \frac{ln ( x )}{( 1 + ln ( x ) ) ^{2}} d x

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

Schritte zur Lösung

\int \frac{ln ( x )}{( 1 + ln ( x ) ) ^{2}} d x

Wende U-Substitution an

= \int \frac{e ^{u} u}{( 1 + u ) ^{2}} d u

Wende die partielle Integration an

= - \frac{e ^{u} u}{1 + u} - \int - e^{u} d u

\int - e^{u} d u = - e^{u}

= - \frac{e ^{u} u}{1 + u} - (- e^{u})

Setze in

u = ln (x)

ein

= - \frac{e ^{l n (x)} ln ( x )}{1 + ln ( x )} - (- e^{l n (x)})

Vereinfache

- \frac{e ^{l n (x)} ln ( x )}{1 + ln ( x )} - (- e^{l n (x)}) : - \frac{x ln ( x )}{1 + ln ( x )} + x

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

Füge eine Konstante zur Lösung hinzu

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

in v erse l a pl a ce \frac{1 + 5 s ^{2}}{s ^{2} ( s - 9 )}

\int 8 e^{5 x} d x

(2 x + y e^{\frac{y}{x}}) d x - x e^{\frac{y}{x}} d y = 0, y (1) = 4

t an g e n t y = x^{2} - 5 x + 5

\int \frac{x ^{2}}{361 - x ^{2}} d x