espandere ln(x+1)+x/(x+1)-1/(x-1)

Soluzione

espandere

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

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

Fasi della soluzione

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

Converti l'elemento in frazione:

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

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

Minimo Comune Multiplo di

x + 1, x - 1, 1 : (x + 1) (x - 1)

Adeguare le frazioni in base al minimo comune multiplo (mcm)

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

Poiché i denominatori sono uguali, combinare le frazioni:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

Espandi

(x + 1) (x - 1) : x^{2} - 1

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

Espandi

x (x - 1) - (x + 1) + ln (x + 1) (x + 1) (x - 1) : x^{2} - 2 x - 1 + x^{2} ln (x + 1) - ln (x + 1)

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

e x p an d \frac{2}{( x + 3 ) ^{2}}

e x p an d 5 + (5 (1 - e)) e^{- x}

e x p an d (- e^{t} cos (t) - sin (t))^{2}

e x p an d \frac{1}{( x + 1 ) ( y + 1 )}

e x p an d \frac{x + 2}{( x - 1 ) ( x ^{2} + x + 1 )}