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受欢迎的微积分 >

integral of (3x-6)e^{x^2-4x}

解答

\int (3 x - 6) e^{x^{2} - 4 x} d x

解答

\frac{3}{2} e^{x (x - 4)} + C

求解步骤

\int (3 x - 6) e^{x^{2} - 4 x} d x

使用换元积分法

= \int \frac{e ^{\frac{u ^{2} - 36}{9}} u}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int e^{\frac{u ^{2} - 36}{9}} u d u

\frac{u ^{2} - 36}{9} = \frac{1}{9} (u^{2} - 36)

= \frac{1}{3} \cdot \int e^{\frac{1}{9} (u^{2} - 36)} u d u

使用换元积分法

= \frac{1}{3} \cdot \int \frac{e ^{\frac{v}{9}}}{2} d v

提出常数:

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

= \frac{1}{3} \cdot \frac{1}{2} \cdot \int e^{\frac{v}{9}} d v

使用换元积分法

= \frac{1}{3} \cdot \frac{1}{2} \cdot \int e^{w} \cdot 9 d w

提出常数:

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

= \frac{1}{3} \cdot \frac{1}{2} \cdot 9 \cdot \int e^{w} d w

使用积分常见定则:

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

= \frac{1}{3} \cdot \frac{1}{2} \cdot 9 e^{w}

代回

= \frac{1}{3} \cdot \frac{1}{2} \cdot 9 e^{\frac{( 3 x - 6 ) ^{2} - 36}{9}}

化简

\frac{1}{3} \cdot \frac{1}{2} \cdot 9 e^{\frac{( 3 x - 6 ) ^{2} - 36}{9}} : \frac{3}{2} e^{x (x - 4)}

= \frac{3}{2} e^{x (x - 4)}

解答补常数

= \frac{3}{2} e^{x (x - 4)} + C

作图

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