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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{x ^{3}}{x + 1} d x

Wende U-Substitution an

= \int \frac{( u - 1 ) ^{3}}{u} d u

Multipliziere aus

\frac{( u - 1 ) ^{3}}{u} : u^{2} - 3 u + 3 - \frac{1}{u}

= \int u^{2} - 3 u + 3 - \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^{2} d u - \int 3 u d u + \int 3 d u - \int \frac{1}{u} d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 3 u d u = \frac{3 u ^{2}}{2}

\int 3 d u = 3 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{3}}{3} - \frac{3 u ^{2}}{2} + 3 u - ln ∣ u ∣

Setze in

u = x + 1

ein

= \frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

integral of 9cos^2(3x)

\int 9 cos^{2} (3 x) d x

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\int (x^{5} + \frac{1}{x ^{3}} + 4) d x

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\int - 3 x^{4} e^{x^{5}} d x

integral of x^{(-1)/3}

\int x^{\frac{- 1}{3}} d x

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\int arccos (\frac{x}{2}) d x