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Beliebt Rechnen >

integral of (2x^{1/2}+3x^2)/(sqrt(x^3))

Lösung

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

Lösung

2 ln ∣ x ∣ + 2 x^{\frac{3}{2}} + C

Schritte zur Lösung

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

x^{3} = x^{\frac{3}{2}},

angenommen

x \geq 0

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

Multipliziere aus

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ + 2 x^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x ∣ + 2 x^{\frac{3}{2}} + C

Graph

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