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Beliebt Rechnen >

integral of (x^3+x+5)/(x^2+2x+2)

Lösung

\int \frac{x ^{3} + x + 5}{x ^{2} + 2 x + 2} d x

Lösung

\frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1) + C

Schritte zur Lösung

\int \frac{x ^{3} + x + 5}{x ^{2} + 2 x + 2} d x

Multipliziere aus

\frac{x ^{3} + x + 5}{x ^{2} + 2 x + 2} : \frac{x ^{3}}{x ^{2} + 2 x + 2} + \frac{x}{x ^{2} + 2 x + 2} + \frac{5}{x ^{2} + 2 x + 2}

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 ^{3}}{x ^{2} + 2 x + 2} d x + \int \frac{x}{x ^{2} + 2 x + 2} d x + \int \frac{5}{x ^{2} + 2 x + 2} d x

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

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

\int \frac{5}{x ^{2} + 2 x + 2} d x = 5 arctan (x + 1)

= \frac{x ^{2}}{2} - 2 x + ln x^{2} + 2 x + 2 + 2 arctan (x + 1) + \frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1) + 5 arctan (x + 1)

Vereinfache

\frac{x ^{2}}{2} - 2 x + ln x^{2} + 2 x + 2 + 2 arctan (x + 1) + \frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1) + 5 arctan (x + 1) : \frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1)

= \frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1)

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1) + C

Graph

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