integral from 1 to 5 of x/(e^x)

解答

\int_{1}^{5} \frac{x}{e ^{x}} d x

- \frac{6}{e ^{5}} + \frac{2}{e}

十进制

0.69533 \dots

求解步骤

\int_{1}^{5} \frac{x}{e ^{x}} d x

使用分布积分法

= [- \frac{x}{e ^{x}} - \int - \frac{1}{e ^{x}} d x]_{1}^{5}

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

= [- \frac{x}{e ^{x}} - \frac{1}{e ^{x}}]_{1}^{5}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= [\frac{- x - 1}{e ^{x}}]_{1}^{5}

带入上下限计算后得:

- \frac{6}{e ^{5}} + \frac{2}{e}

= - \frac{6}{e ^{5}} + \frac{2}{e}

t an g e n t x^{2} - 6 x

f (x) = \frac{3}{x - 2}

l a pl a ce t r an s f or m - 4 cos (3 t)

\int \frac{10800 q}{9 q ^{2} + 675} d q

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