integral of x/(e^{\frac{x^2){2}}}

解答

\int \frac{x}{e ^{\frac{x ^{2}}{2}}} d x

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

求解步骤

\int \frac{x}{e ^{\frac{x ^{2}}{2}}} d x

使用换元积分法

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

使用分布积分法

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

\int - \frac{1}{2} e^{- \frac{u}{2}} u d u = e^{- \frac{u}{2}} (u + 2)

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

u = x^{2}

代回

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

化简

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

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

解答补常数

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

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