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

解

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

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

解答ステップ

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

拡張

\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

総和規則を適用する:

\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 ∣

定数を解答に追加する

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