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受欢迎的微积分 >

integral of 3x^5e^{x^3}

解答

\int 3 x^{5} e^{x^{3}} d x

解答

e^{x^{3}} x^{3} - e^{x^{3}} + C

求解步骤

\int 3 x^{5} e^{x^{3}} d x

提出常数:

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

= 3 \cdot \int x^{5} e^{x^{3}} d x

使用换元积分法

= 3 \cdot \int \frac{e ^{u^{\frac{3}{5}}} u ^{\frac{1}{5}}}{5} d u

提出常数:

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

= 3 \cdot \frac{1}{5} \cdot \int e^{u^{\frac{3}{5}}} u^{\frac{1}{5}} d u

使用换元积分法

= 3 \cdot \frac{1}{5} \cdot \int \frac{5 e ^{v} v}{3} d v

提出常数:

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

= 3 \cdot \frac{1}{5} \cdot \frac{5}{3} \cdot \int e^{v} v d v

使用分布积分法

= 3 \cdot \frac{1}{5} \cdot \frac{5}{3} (e^{v} v - \int e^{v} d v)

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

= 3 \cdot \frac{1}{5} \cdot \frac{5}{3} (e^{v} v - e^{v})

代回

= 3 \cdot \frac{1}{5} \cdot \frac{5}{3} (e^{(x^{5})^{\frac{3}{5}}} (x^{5})^{\frac{3}{5}} - e^{(x^{5})^{\frac{3}{5}}})

化简

3 \cdot \frac{1}{5} \cdot \frac{5}{3} (e^{(x^{5})^{\frac{3}{5}}} (x^{5})^{\frac{3}{5}} - e^{(x^{5})^{\frac{3}{5}}}) : e^{x^{3}} x^{3} - e^{x^{3}}

= e^{x^{3}} x^{3} - e^{x^{3}}

解答补常数

= e^{x^{3}} x^{3} - e^{x^{3}} + C

作图

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