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受欢迎的微积分 >

integral of (5e^{x^{-2}})/(x^3)

解答

\int \frac{5 e ^{x^{- 2}}}{x ^{3}} d x

解答

- \frac{5}{2} e^{\frac{1}{x ^{2}}} + C

求解步骤

\int \frac{5 e ^{x^{- 2}}}{x ^{3}} d x

提出常数:

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

= 5 \cdot \int \frac{e ^{x^{- 2}}}{x ^{3}} d x

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

= 5 \cdot \int \frac{e ^{\frac{1}{x ^{2}}}}{x ^{3}} d x

使用换元积分法

= 5 \cdot \int \frac{e ^{\frac{1}{u}}}{2 u ^{2}} d u

提出常数:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{e ^{\frac{1}{u}}}{u ^{2}} d u

使用换元积分法

= 5 \cdot \frac{1}{2} \cdot \int - e^{v} d v

提出常数:

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

= 5 \cdot \frac{1}{2} (- \int e^{v} d v)

使用积分常见定则:

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

= 5 \cdot \frac{1}{2} (- e^{v})

代回

= 5 \cdot \frac{1}{2} (- e^{\frac{1}{x ^{2}}})

化简

5 \cdot \frac{1}{2} (- e^{\frac{1}{x ^{2}}}) : - \frac{5}{2} e^{\frac{1}{x ^{2}}}

= - \frac{5}{2} e^{\frac{1}{x ^{2}}}

解答补常数

= - \frac{5}{2} e^{\frac{1}{x ^{2}}} + C

作图

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