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受欢迎的微积分 >

integral from 0 to 1 of x^4e^{3x^5}

解答

\int_{0}^{1} x^{4} e^{3 x^{5}} d x

解答

\frac{e ^{3} - 1}{15}

+1

十进制

1.27236 \dots

求解步骤

\int_{0}^{1} x^{4} e^{3 x^{5}} d x

使用换元积分法

= \int_{0}^{1} \frac{e ^{3 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

= \frac{1}{4} \cdot \int_{0}^{1} e^{3 u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

使用换元积分法

= \frac{1}{4} \cdot \int_{0}^{1} \frac{4 e ^{3 v}}{5} d v

提出常数:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \int_{0}^{1} e^{3 v} d v

使用换元积分法

= \frac{1}{4} \cdot \frac{4}{5} \cdot \int_{0}^{3} e^{w} \frac{1}{3} d w

提出常数:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{3} \cdot \int_{0}^{3} e^{w} d w

使用积分常见定则:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{3} [e^{w}]_{0}^{3}

化简

\frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{3} [e^{w}]_{0}^{3} : \frac{1}{15} [e^{w}]_{0}^{3}

= \frac{1}{15} [e^{w}]_{0}^{3}

带入上下限计算后得:

e^{3} - 1

= \frac{1}{15} (e^{3} - 1)

化简

= \frac{e ^{3} - 1}{15}

作图

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