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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{4} (2 ln ∣ x ∣ - \frac{6}{x}) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vereinfache

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

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

Multipliziere aus

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{3}{x ^{\frac{3}{2}}} d x = - \frac{6}{x}

= \frac{1}{4} (2 ln ∣ x ∣ - \frac{6}{x})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (2 ln ∣ x ∣ - \frac{6}{x}) + C

Graph

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