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受欢迎的微积分 >

integral of e^{x^3}(3x^2)

解答

\int e^{x^{3}} (3 x^{2}) d x

解答

e^{x^{3}} + C

求解步骤

\int e^{x^{3}} \cdot 3 x^{2} d x

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= 3 \cdot \frac{1}{3} e^{u}

u = x^{3}

代回

= 3 \cdot \frac{1}{3} e^{x^{3}}

化简

3 \cdot \frac{1}{3} e^{x^{3}} : e^{x^{3}}

= e^{x^{3}}

解答补常数

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

作图

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