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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u ^{2} - 2 u + 2}{u} d u

Multipliziere aus

\frac{u ^{2} - 2 u + 2}{u} : u - 2 + \frac{2}{u}

= \int u - 2 + \frac{2}{u} d u

Wende die Summenregel an:

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

= \int u d u - \int 2 d u + \int \frac{2}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + 2 ln ∣ u ∣

Setze in

u = 1 + x

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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