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

解答

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

- \frac{x ^{2}}{e ^{x}} - \frac{2 x}{e ^{x}} - \frac{4}{e ^{x}} + C

求解步骤

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

乘开

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

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

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= - \frac{x ^{2}}{e ^{x}} + 2 (- \frac{x}{e ^{x}} - \frac{1}{e ^{x}}) - \frac{2}{e ^{x}}

化简

- \frac{x ^{2}}{e ^{x}} + 2 (- \frac{x}{e ^{x}} - \frac{1}{e ^{x}}) - \frac{2}{e ^{x}} : - \frac{x ^{2}}{e ^{x}} - \frac{2 x}{e ^{x}} - \frac{4}{e ^{x}}

= - \frac{x ^{2}}{e ^{x}} - \frac{2 x}{e ^{x}} - \frac{4}{e ^{x}}

解答补常数

= - \frac{x ^{2}}{e ^{x}} - \frac{2 x}{e ^{x}} - \frac{4}{e ^{x}} + C

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