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受欢迎的微积分 >

integral of e^{2x+x^2}

解答

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

解答

\frac{π}{2 e} erfi (x + 1) + C

求解步骤

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

对

2 x + x^{2}

配方:

(x + 1)^{2} - 1

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

使用换元积分法

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

化简

e^{u^{2} - 1} : e^{u^{2}} e^{- 1}

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

提出常数:

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

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

使用指数法则:

a^{- 1} = \frac{1}{a}

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

这是非初等函数积分:

\int e^{u^{2}} d u = \frac{π}{2} erfi (u)

= \frac{1}{e} \cdot \frac{π}{2} erfi (u)

u = x + 1

代回

= \frac{1}{e} \cdot \frac{π}{2} erfi (x + 1)

化简

\frac{1}{e} \cdot \frac{π}{2} erfi (x + 1) : \frac{π}{2 e} erfi (x + 1)

= \frac{π}{2 e} erfi (x + 1)

解答补常数

= \frac{π}{2 e} erfi (x + 1) + C

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