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受欢迎的微积分 >

integral of 5x^4e^{9x^5}

解答

\int 5 x^{4} e^{9 x^{5}} d x

解答

\frac{1}{9} e^{9 x^{5}} + C

求解步骤

\int 5 x^{4} e^{9 x^{5}} d x

提出常数:

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

= 5 \cdot \int x^{4} e^{9 x^{5}} d x

使用换元积分法

= 5 \cdot \int \frac{e ^{9 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

= 5 \cdot \frac{1}{4} \cdot \int e^{9 u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

使用换元积分法

= 5 \cdot \frac{1}{4} \cdot \int \frac{4 e ^{9 v}}{5} d v

提出常数:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{9 v} d v

使用换元积分法

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{w} \frac{1}{9} d w

提出常数:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{9} \cdot \int e^{w} d w

使用积分常见定则:

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

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{9} e^{w}

代回

= 5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{9} e^{9 (x^{4})^{\frac{5}{4}}}

化简

5 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{9} e^{9 (x^{4})^{\frac{5}{4}}} : \frac{1}{9} e^{9 x^{5}}

= \frac{1}{9} e^{9 x^{5}}

解答补常数

= \frac{1}{9} e^{9 x^{5}} + C

作图

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