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Beliebt Rechnen >

integral of (x^3+x+1)/(x^2+4)

Lösung

\int \frac{x ^{3} + x + 1}{x ^{2} + 4} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{3} + x + 1}{x ^{2} + 4} d x

Multipliziere aus

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

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} + 4} d x + \int \frac{x}{x ^{2} + 4} d x + \int \frac{1}{x ^{2} + 4} d x

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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