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Beliebt Rechnen >

integral of 1/((x-1)(x+1)^3)

Lösung

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

Lösung

\frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{2}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x - 1 ) ( x + 1 ) ^{3}} : \frac{1}{8 ( x - 1 )} - \frac{1}{8 ( x + 1 )} - \frac{1}{4 ( x + 1 ) ^{2}} - \frac{1}{2 ( x + 1 ) ^{3}}

= \int \frac{1}{8 ( x - 1 )} - \frac{1}{8 ( x + 1 )} - \frac{1}{4 ( x + 1 ) ^{2}} - \frac{1}{2 ( x + 1 ) ^{3}} 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}{8 ( x - 1 )} d x - \int \frac{1}{8 ( x + 1 )} d x - \int \frac{1}{4 ( x + 1 ) ^{2}} d x - \int \frac{1}{2 ( x + 1 ) ^{3}} d x

\int \frac{1}{8 ( x - 1 )} d x = \frac{1}{8} ln ∣ x - 1 ∣

\int \frac{1}{8 ( x + 1 )} d x = \frac{1}{8} ln ∣ x + 1 ∣

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

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

= \frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 1 ∣ - (- \frac{1}{4 ( x + 1 )}) - (- \frac{1}{4 ( x + 1 ) ^{2}})

Vereinfache

= \frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{2}} + C

Graph

Beliebte Beispiele

integral of (1+1/u+1/(u^2))

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

integral of (x+3)/(x^2+2x+5)

\int \frac{x + 3}{x ^{2} + 2 x + 5} d x

integral of 1/((x+7)(49x^2+7x+1))

\int \frac{1}{( x + 7 ) ( 49 x ^{2} + 7 x + 1 )} d x

integral of 1/(x(1+x^4))

\int \frac{1}{x ( 1 + x ^{4} )} d x

integral of 2^{1-2x}

\int 2^{1 - 2 x} d x