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受欢迎的微积分 >

integral of 2x^5e^{x^6}

解答

\int 2 x^{5} e^{x^{6}} d x

解答

\frac{1}{3} e^{x^{6}} + C

求解步骤

\int 2 x^{5} e^{x^{6}} d x

提出常数:

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

= 2 \cdot \int x^{5} e^{x^{6}} d x

使用换元积分法

= 2 \cdot \int \frac{e ^{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

= 2 \cdot \frac{1}{5} \cdot \int e^{u^{\frac{6}{5}}} u^{\frac{1}{5}} d u

使用换元积分法

= 2 \cdot \frac{1}{5} \cdot \int \frac{5 e ^{v}}{6} d v

提出常数:

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

= 2 \cdot \frac{1}{5} \cdot \frac{5}{6} \cdot \int e^{v} d v

使用积分常见定则:

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

= 2 \cdot \frac{1}{5} \cdot \frac{5}{6} e^{v}

代回

= 2 \cdot \frac{1}{5} \cdot \frac{5}{6} e^{(x^{5})^{\frac{6}{5}}}

化简

2 \cdot \frac{1}{5} \cdot \frac{5}{6} e^{(x^{5})^{\frac{6}{5}}} : \frac{1}{3} e^{x^{6}}

= \frac{1}{3} e^{x^{6}}

解答补常数

= \frac{1}{3} e^{x^{6}} + C

作图

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