integral of (sqrt(x)-x^3*e^x+x^2)/(x^3)

Lösung

\int \frac{x - x ^{3} \cdot e ^{x} + x ^{2}}{x ^{3}} d x

- \frac{2}{3 x ^{\frac{3}{2}}} - e^{x} + ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x - x ^{3} e ^{x} + x ^{2}}{x ^{3}} d x

Multipliziere aus

\frac{x - x ^{3} e ^{x} + x ^{2}}{x ^{3}} : \frac{1}{x ^{\frac{5}{2}}} - e^{x} + \frac{1}{x}

= \int \frac{1}{x ^{\frac{5}{2}}} - e^{x} + \frac{1}{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

= \int \frac{1}{x ^{\frac{5}{2}}} d x - \int e^{x} d x + \int \frac{1}{x} d x

\int \frac{1}{x ^{\frac{5}{2}}} d x = - \frac{2}{3 x ^{\frac{3}{2}}}

\int e^{x} d x = e^{x}

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{2}{3 x ^{\frac{3}{2}}} - e^{x} + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{3 x ^{\frac{3}{2}}} - e^{x} + ln ∣ x ∣ + C

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