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受欢迎的微积分 >

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

解答

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

解答

\frac{e ^{81} - 1}{4}

+1

十进制

3.76524 E 34

求解步骤

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

使用换元积分法

= \int_{0}^{27} \frac{e ^{u^{\frac{4}{3}}} u ^{\frac{1}{3}}}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int_{0}^{27} e^{u^{\frac{4}{3}}} u^{\frac{1}{3}} d u

使用换元积分法

= \frac{1}{3} \cdot \int_{0}^{81} \frac{3 e ^{v}}{4} d v

提出常数:

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

= \frac{1}{3} \cdot \frac{3}{4} \cdot \int_{0}^{81} e^{v} d v

使用积分常见定则:

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

= \frac{1}{3} \cdot \frac{3}{4} [e^{v}]_{0}^{81}

化简

\frac{1}{3} \cdot \frac{3}{4} [e^{v}]_{0}^{81} : \frac{1}{4} [e^{v}]_{0}^{81}

= \frac{1}{4} [e^{v}]_{0}^{81}

带入上下限计算后得:

e^{81} - 1

= \frac{1}{4} (e^{81} - 1)

化简

= \frac{e ^{81} - 1}{4}

作图

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