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受欢迎的微积分 >

integral of (7x^2)e^{x^3+1}

解答

\int (7 x^{2}) e^{x^{3} + 1} d x

解答

\frac{7}{3} e^{x^{3} + 1} + C

求解步骤

\int 7 x^{2} e^{x^{3} + 1} d x

提出常数:

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

= 7 \cdot \int x^{2} e^{x^{3} + 1} d x

使用换元积分法

= 7 \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

= 7 \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}

= 7 \cdot \frac{1}{3} e^{u}

u = x^{3} + 1

代回

= 7 \cdot \frac{1}{3} e^{x^{3} + 1}

化简

7 \cdot \frac{1}{3} e^{x^{3} + 1} : \frac{7}{3} e^{x^{3} + 1}

= \frac{7}{3} e^{x^{3} + 1}

解答补常数

= \frac{7}{3} e^{x^{3} + 1} + C

作图

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