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受欢迎的微积分 >

integral of 6x^4e^{x^5}

解答

\int 6 x^{4} e^{x^{5}} d x

解答

\frac{6}{5} e^{x^{5}} + C

求解步骤

\int 6 x^{4} e^{x^{5}} d x

提出常数:

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

= 6 \cdot \int x^{4} e^{x^{5}} d x

使用换元积分法

= 6 \cdot \int \frac{e ^{u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

提出常数:

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

= 6 \cdot \frac{1}{4} \cdot \int e^{u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

使用换元积分法

= 6 \cdot \frac{1}{4} \cdot \int \frac{4 e ^{v}}{5} d v

提出常数:

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

= 6 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{v} d v

使用积分常见定则:

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

= 6 \cdot \frac{1}{4} \cdot \frac{4}{5} e^{v}

代回

= 6 \cdot \frac{1}{4} \cdot \frac{4}{5} e^{(x^{4})^{\frac{5}{4}}}

化简

6 \cdot \frac{1}{4} \cdot \frac{4}{5} e^{(x^{4})^{\frac{5}{4}}} : \frac{6}{5} e^{x^{5}}

= \frac{6}{5} e^{x^{5}}

解答补常数

= \frac{6}{5} e^{x^{5}} + C

作图

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