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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{2}}{4} + ln ∣ x ∣ + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Multipliziere aus

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

= \frac{1}{2} \cdot \int x + \frac{2}{x} 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

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

\int x d x = \frac{x ^{2}}{2}

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

Vereinfache

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

= \frac{x ^{2}}{4} + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{4} + ln ∣ x ∣ + C

Graph

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