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受欢迎的微积分 >

integral from 0 to 1 of e^{-2x}

解答

\int_{0}^{1} e^{- 2 x} d x

解答

\frac{e ^{2} - 1}{2 e ^{2}}

+1

十进制

0.43233 \dots

求解步骤

\int_{0}^{1} e^{- 2 x} d x

使用换元积分法

= \int_{0}^{- 2} - \frac{1}{2} e^{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 2}^{0} - \frac{1}{2} e^{u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \frac{1}{2} \cdot \int_{- 2}^{0} e^{u} d u)

使用积分常见定则:

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

= - (- \frac{1}{2} [e^{u}]_{- 2}^{0})

化简

= \frac{1}{2} [e^{u}]_{- 2}^{0}

带入上下限计算后得:

1 - \frac{1}{e ^{2}}

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

化简

= \frac{e ^{2} - 1}{2 e ^{2}}

作图

流行的例子

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a re a y = x, x = 4

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\int tan^{7} (x) d x

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d er i v a t i v e y = (x^{5} + 1)^{6}

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d er i v a t i v e y = 4 x^{2} + 2

integral of 1/((1+x^2))

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