integral of e^{-ax^2}

解答

\int e^{- a x^{2}} d x

\frac{π}{2 a} erf (a x) + C

求解步骤

\int e^{- a x^{2}} d x

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

使用换元积分法

= \int \frac{e ^{- u^{2}}}{a} d u

提出常数:

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

= \frac{1}{a} \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} erf (u)

= \frac{1}{a} \cdot \frac{π}{2} erf (u)

u = a x

代回

= \frac{1}{a} \cdot \frac{π}{2} erf (a x)

化简

\frac{1}{a} \cdot \frac{π}{2} erf (a x) : \frac{π}{2 a} erf (a x)

= \frac{π}{2 a} erf (a x)

解答补常数

= \frac{π}{2 a} erf (a x) + C

d er i v a t i v e y = 2 3 x

t an g e n t y = (1 + 4 x)^{11}, (0, 1)

t an g e n t f (x) = \frac{x}{x + 6}, at x = 4

d er i v a t i v e e^{\frac{z}{z - 6}}

\int \frac{16 s - 4}{3 - 2 s + 4 s ^{2}} d s