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Beliebt Rechnen >

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

Lösung

integral

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

= \frac{2}{- x + 1} + ln ∣ - x + 1 ∣

Füge eine Konstante zur Lösung hinzu

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

Graph

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