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

Lösung

integral

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

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

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

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

Setze in

u = x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 2^{x} \cdot e^{2} x

in t e g r a l e^{x} \cdot cos^{2} (e^{x})

in t e g r a l \frac{e ^{- 1}}{x ^{3}}

in t e g r a l \frac{r ^{2}}{( 1 + r ^{2} ) ^{2}}

in t e g r a l \frac{( 4 + 12 x )}{4}