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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

= \int u - 2 + \frac{1}{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{1}{u} d u

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + ln ∣ u ∣

Setze in

u = 1 + t

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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