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受欢迎的微积分 >

integral from 0 to 3 of x^7e^{x^4}

解答

\int_{0}^{3} x^{7} e^{x^{4}} d x

解答

\frac{80 e ^{81} + 1}{4}

+1

十进制

3.01219 E 36

求解步骤

\int_{0}^{3} x^{7} e^{x^{4}} d x

使用换元积分法

= \int_{0}^{2187} \frac{e ^{u^{\frac{4}{7}}} u ^{\frac{1}{7}}}{7} d u

提出常数:

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

= \frac{1}{7} \cdot \int_{0}^{2187} e^{u^{\frac{4}{7}}} u^{\frac{1}{7}} d u

使用换元积分法

= \frac{1}{7} \cdot \int_{0}^{81} \frac{7 e ^{v} v}{4} d v

提出常数:

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

= \frac{1}{7} \cdot \frac{7}{4} \cdot \int_{0}^{81} e^{v} v d v

使用分布积分法

= \frac{1}{7} \cdot \frac{7}{4} [e^{v} v - \int e^{v} d v]_{0}^{81}

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

= \frac{1}{7} \cdot \frac{7}{4} [e^{v} v - e^{v}]_{0}^{81}

化简

\frac{1}{7} \cdot \frac{7}{4} [e^{v} v - e^{v}]_{0}^{81} : \frac{1}{4} [e^{v} v - e^{v}]_{0}^{81}

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

带入上下限计算后得:

80 e^{81} + 1

= \frac{1}{4} (80 e^{81} + 1)

化简

= \frac{80 e ^{81} + 1}{4}

作图

流行的例子

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