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受欢迎的微积分 >

integral of 6x^5e^{8x^6}

解答

\int 6 x^{5} e^{8 x^{6}} d x

解答

\frac{1}{8} e^{8 x^{6}} + C

求解步骤

\int 6 x^{5} e^{8 x^{6}} d x

提出常数:

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

= 6 \cdot \int x^{5} e^{8 x^{6}} d x

使用换元积分法

= 6 \cdot \int \frac{e ^{8 u^{\frac{6}{5}}} u ^{\frac{1}{5}}}{5} d u

提出常数:

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

= 6 \cdot \frac{1}{5} \cdot \int e^{8 u^{\frac{6}{5}}} u^{\frac{1}{5}} d u

使用换元积分法

= 6 \cdot \frac{1}{5} \cdot \int \frac{5 e ^{8 v}}{6} d v

提出常数:

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \int e^{8 v} d v

使用换元积分法

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \int e^{w} \frac{1}{8} d w

提出常数:

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{8} \cdot \int e^{w} d w

使用积分常见定则:

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

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{8} e^{w}

代回

= 6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{8} e^{8 (x^{5})^{\frac{6}{5}}}

化简

6 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \frac{1}{8} e^{8 (x^{5})^{\frac{6}{5}}} : \frac{1}{8} e^{8 x^{6}}

= \frac{1}{8} e^{8 x^{6}}

解答补常数

= \frac{1}{8} e^{8 x^{6}} + C

作图

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