integral from 1 to 2.2 of x^3*e^{x+1}

解答

\int_{1}^{2.2} x^{3} \cdot e^{x + 1} d x

96.42237 \dots

求解步骤

\int_{1}^{2.2} x^{3} e^{x + 1} d x

x^{3} e^{x + 1} = e^{1} x^{3} e^{x}

= \int_{1}^{2.2} e^{1} x^{3} e^{x} d x

提出常数:

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

= e^{1} \cdot \int_{1}^{2.2} x^{3} e^{x} d x

化简

= e \cdot \int_{1}^{2.2} x^{3} e^{x} d x

使用分布积分法

= e [x^{3} e^{x} - \int 3 x^{2} e^{x} d x]_{1}^{2.2}

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

= e [x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))]_{1}^{2.2}

带入上下限计算后得:

35.47180 \dots

= e 35.47180 \dots

化简

= 96.42237 \dots

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