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受欢迎的微积分 >

integral of-4x^3e^{x^4}

解答

\int - 4 x^{3} e^{x^{4}} d x

解答

- e^{x^{4}} + C

求解步骤

\int - 4 x^{3} e^{x^{4}} d x

提出常数:

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

= - 4 \cdot \int x^{3} e^{x^{4}} d x

使用换元积分法

= - 4 \cdot \int \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

= - 4 \cdot \frac{1}{3} \cdot \int e^{u^{\frac{4}{3}}} u^{\frac{1}{3}} d u

使用换元积分法

= - 4 \cdot \frac{1}{3} \cdot \int \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

= - 4 \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \int e^{v} d v

使用积分常见定则:

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

= - 4 \cdot \frac{1}{3} \cdot \frac{3}{4} e^{v}

代回

= - 4 \cdot \frac{1}{3} \cdot \frac{3}{4} e^{(x^{3})^{\frac{4}{3}}}

化简

- 4 \cdot \frac{1}{3} \cdot \frac{3}{4} e^{(x^{3})^{\frac{4}{3}}} : - e^{x^{4}}

= - e^{x^{4}}

解答补常数

= - e^{x^{4}} + C

作图

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