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受欢迎的微积分 >

integral of x^6e^{x^7}

解答

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

解答

\frac{1}{7} e^{x^{7}} + C

求解步骤

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

使用换元积分法

= \int \frac{e ^{u^{\frac{7}{6}}} u ^{\frac{1}{6}}}{6} d u

提出常数:

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

= \frac{1}{6} \cdot \int e^{u^{\frac{7}{6}}} u^{\frac{1}{6}} d u

使用换元积分法

= \frac{1}{6} \cdot \int \frac{6 e ^{v}}{7} d v

提出常数:

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

= \frac{1}{6} \cdot \frac{6}{7} \cdot \int e^{v} d v

使用积分常见定则:

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

= \frac{1}{6} \cdot \frac{6}{7} e^{v}

代回

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

化简

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

= \frac{1}{7} e^{x^{7}}

解答补常数

= \frac{1}{7} e^{x^{7}} + C

作图

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