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Beliebt Rechnen >

integral of (x^3)/((x+1)^4)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{3}}{( x + 1 ) ^{4}} : \frac{1}{x + 1} - \frac{3}{( x + 1 ) ^{2}} + \frac{3}{( x + 1 ) ^{3}} - \frac{1}{( x + 1 ) ^{4}}

= \int \frac{1}{x + 1} - \frac{3}{( x + 1 ) ^{2}} + \frac{3}{( x + 1 ) ^{3}} - \frac{1}{( x + 1 ) ^{4}} d x

Wende die Summenregel an:

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

= \int \frac{1}{x + 1} d x - \int \frac{3}{( x + 1 ) ^{2}} d x + \int \frac{3}{( x + 1 ) ^{3}} d x - \int \frac{1}{( x + 1 ) ^{4}} d x

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

\int \frac{3}{( x + 1 ) ^{2}} d x = - \frac{3}{x + 1}

\int \frac{3}{( x + 1 ) ^{3}} d x = - \frac{3}{2 ( x + 1 ) ^{2}}

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of sin^5(x)*cos^6(x)

\int sin^{5} (x) \cdot cos^{6} (x) d x

integral of (x-sqrt(2x))/(x^3)

\int \frac{x - 2 x}{x ^{3}} d x

integral of (3x+4)/(x^3-2x-4)

\int \frac{3 x + 4}{x ^{3} - 2 x - 4} d x

integral of (x^2+\sqrt[4]{x})

\int (x^{2} + 4 x) d x

integral of (2t^2+1)

\int (2 t^{2} + 1) d t