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受欢迎的微积分 >

integral of-7x^3e^{x^2}

解答

\int - 7 x^{3} e^{x^{2}} d x

解答

- \frac{7}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

求解步骤

\int - 7 x^{3} e^{x^{2}} d x

提出常数:

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

= - 7 \cdot \int x^{3} e^{x^{2}} d x

使用换元积分法

= - 7 \cdot \int \frac{e ^{u^{\frac{2}{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

= - 7 \cdot \frac{1}{3} \cdot \int e^{u^{\frac{2}{3}}} u^{\frac{1}{3}} d u

使用换元积分法

= - 7 \cdot \frac{1}{3} \cdot \int \frac{3 e ^{v} v}{2} d v

提出常数:

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

= - 7 \cdot \frac{1}{3} \cdot \frac{3}{2} \cdot \int e^{v} v d v

使用分布积分法

= - 7 \cdot \frac{1}{3} \cdot \frac{3}{2} (e^{v} v - \int e^{v} d v)

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

= - 7 \cdot \frac{1}{3} \cdot \frac{3}{2} (e^{v} v - e^{v})

代回

= - 7 \cdot \frac{1}{3} \cdot \frac{3}{2} (e^{(x^{3})^{\frac{2}{3}}} (x^{3})^{\frac{2}{3}} - e^{(x^{3})^{\frac{2}{3}}})

化简

- 7 \cdot \frac{1}{3} \cdot \frac{3}{2} (e^{(x^{3})^{\frac{2}{3}}} (x^{3})^{\frac{2}{3}} - e^{(x^{3})^{\frac{2}{3}}}) : - \frac{7}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

= - \frac{7}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

解答补常数

= - \frac{7}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + C

作图

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