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

解

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

- \frac{2}{x} - ln ∣ x ∣ + e^{x} + C

解答ステップ

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

拡張

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= - \frac{2}{x} - ln ∣ x ∣ + e^{x}

定数を解答に追加する

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