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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

= x - ln ∣ x + 1 ∣ - \frac{1}{x + 1}

Füge eine Konstante zur Lösung hinzu

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

Graph

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