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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{2} ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

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

Vereinfache

= \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{2} ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{x + 1} + \frac{1}{2} ln ∣ x - 1 ∣ + C

Graph

Beliebte Beispiele

integral of (ln^4(x))/(x(ln^2(x)-1))

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

integral of (sin(2x))/(30+cos^2(x))

\int \frac{sin ( 2 x )}{30 + cos ^{2} ( x )} d x

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integral of (y^2+5)/(y^2+9)

\int \frac{y ^{2} + 5}{y ^{2} + 9} d y