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受欢迎的微积分 >

integral of 4x^2e^{x^3}

解答

\int 4 x^{2} e^{x^{3}} d x

解答

\frac{4}{3} e^{x^{3}} + C

求解步骤

\int 4 x^{2} e^{x^{3}} d x

提出常数:

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

= 4 \cdot \int x^{2} e^{x^{3}} d x

使用换元积分法

= 4 \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

= 4 \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}

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

u = x^{3}

代回

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

化简

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

= \frac{4}{3} e^{x^{3}}

解答补常数

= \frac{4}{3} e^{x^{3}} + C

作图

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