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Beliebt Rechnen >

integral of (x^4+x^3+x+5)/(x+1)

Lösung

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

Lösung

\frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + 3 x + 4 ln ∣ x + 1 ∣ + \frac{1}{3} (x + 1)^{3} - \frac{3}{2} (x + 1)^{2} + 4 + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{4} + x ^{3} + x + 5}{x + 1} : \frac{x ^{4}}{x + 1} + \frac{x ^{3}}{x + 1} + \frac{x}{x + 1} + \frac{5}{x + 1}

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

\int \frac{x ^{4}}{x + 1} d x = \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣

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

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

\int \frac{5}{x + 1} d x = 5 ln ∣ x + 1 ∣

= \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣ + \frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣ + x + 1 - ln ∣ x + 1 ∣ + 5 ln ∣ x + 1 ∣

Vereinfache

\frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x + 1 ∣ + \frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣ + x + 1 - ln ∣ x + 1 ∣ + 5 ln ∣ x + 1 ∣ : \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + 3 x + 4 ln ∣ x + 1 ∣ + \frac{1}{3} (x + 1)^{3} - \frac{3}{2} (x + 1)^{2} + 4

= \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + 3 x + 4 ln ∣ x + 1 ∣ + \frac{1}{3} (x + 1)^{3} - \frac{3}{2} (x + 1)^{2} + 4

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{4}}{4} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + 3 x + 4 ln ∣ x + 1 ∣ + \frac{1}{3} (x + 1)^{3} - \frac{3}{2} (x + 1)^{2} + 4 + C

Graph

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