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受欢迎的微积分 >

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

解答

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

解答

\frac{e ^{4} - 1}{2 e ^{2}}

+1

十进制

3.62686 \dots

求解步骤

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

使用换元积分法

= \int_{2}^{- 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}^{2} - \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}^{2} 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}^{2})

化简

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

带入上下限计算后得:

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

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

化简

= \frac{e ^{4} - 1}{2 e ^{2}}

作图

流行的例子

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

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

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\int_{- 1}^{5} (3 x^{4} + 2 x^{2} + 3) d x

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\int_{0}^{5} - 3 x d x

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\int_{- 2}^{5} 1 d x

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\int_{1}^{2} (2 x + 3) d x