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受欢迎的微积分 >

integral of 7xe^{6x}

解答

\int 7 x e^{6 x} d x

解答

\frac{7}{36} (6 e^{6 x} x - e^{6 x}) + C

求解步骤

\int 7 x e^{6 x} d x

提出常数:

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

= 7 \cdot \int x e^{6 x} d x

使用换元积分法

= 7 \cdot \int \frac{e ^{u} u}{36} d u

提出常数:

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

= 7 \cdot \frac{1}{36} \cdot \int e^{u} u d u

使用分布积分法

= 7 \cdot \frac{1}{36} (e^{u} u - \int e^{u} d u)

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

= 7 \cdot \frac{1}{36} (e^{u} u - e^{u})

u = 6 x

代回

= 7 \cdot \frac{1}{36} (e^{6 x} \cdot 6 x - e^{6 x})

化简

7 \cdot \frac{1}{36} (e^{6 x} \cdot 6 x - e^{6 x}) : \frac{7}{36} (6 e^{6 x} x - e^{6 x})

= \frac{7}{36} (6 e^{6 x} x - e^{6 x})

解答补常数

= \frac{7}{36} (6 e^{6 x} x - e^{6 x}) + C

作图

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