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integral from 0 to 2 of (4x+5)e^{-x}

解答

\int_{0}^{2} (4 x + 5) e^{- x} d x

解答

- \frac{17}{e ^{2}} + 9

+1

十进制

6.69930 \dots

求解步骤

\int_{0}^{2} (4 x + 5) e^{- x} d x

乘以

4 x + 5

的共轭根式

: \frac{16 x ^{2} - 25}{4 x - 5}

= \int_{0}^{2} \frac{16 x ^{2} - 25}{4 x - 5} e^{- x} d x

找到无定义的点（奇点）:

x = \frac{5}{4}

如果存在

b, a < b < c, f (b) =

无定义

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{\frac{5}{4}} \frac{16 x ^{2} - 25}{4 x - 5} e^{- x} d x + \int_{\frac{5}{4}}^{2} \frac{16 x ^{2} - 25}{4 x - 5} e^{- x} d x

\int_{0}^{\frac{5}{4}} \frac{16 x ^{2} - 25}{4 x - 5} e^{- x} d x = - \frac{14}{e ^{\frac{5}{4}}} + 9

\int_{\frac{5}{4}}^{2} \frac{16 x ^{2} - 25}{4 x - 5} e^{- x} d x = - \frac{17 e ^{\frac{5}{4}} - 14 e ^{2}}{e ^{\frac{13}{4}}}

= - \frac{14}{e ^{\frac{5}{4}}} + 9 - \frac{17 e ^{\frac{5}{4}} - 14 e ^{2}}{e ^{\frac{13}{4}}}

化简

- \frac{14}{e ^{\frac{5}{4}}} + 9 - \frac{17 e ^{\frac{5}{4}} - 14 e ^{2}}{e ^{\frac{13}{4}}} : - \frac{17}{e ^{2}} + 9

= - \frac{17}{e ^{2}} + 9

作图

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