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受欢迎的微积分 >

integral of 3e^{2x^3+3x^2-9}(x^2+x)

解答

\int 3 e^{2 x^{3} + 3 x^{2} - 9} (x^{2} + x) d x

解答

\frac{1}{2} e^{2 x^{3} + 3 x^{2} - 9} + C

求解步骤

\int 3 e^{2 x^{3} + 3 x^{2} - 9} (x^{2} + x) d x

提出常数:

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

= 3 \cdot \int e^{2 x^{3} + 3 x^{2} - 9} (x^{2} + x) d x

使用换元积分法

= 3 \cdot \int \frac{e ^{u}}{6} d u

提出常数:

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

= 3 \cdot \frac{1}{6} \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}{6} e^{u}

u = 2 x^{3} + 3 x^{2} - 9

代回

= 3 \cdot \frac{1}{6} e^{2 x^{3} + 3 x^{2} - 9}

化简

3 \cdot \frac{1}{6} e^{2 x^{3} + 3 x^{2} - 9} : \frac{1}{2} e^{2 x^{3} + 3 x^{2} - 9}

= \frac{1}{2} e^{2 x^{3} + 3 x^{2} - 9}

解答补常数

= \frac{1}{2} e^{2 x^{3} + 3 x^{2} - 9} + C

作图

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