de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x^{1.4}+9x^{3.5}

\int x^{1.4} + 9 x^{3.5} d x

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

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

integral of sec^2(5x+3)

\int sec^{2} (5 x + 3) d x

integral of 10xe^{-x}

\int 10 x e^{- x} d x

integral of p(p+5)^6

\int p (p + 5)^{6} d p