integral from 0 to pi/2 of 5e^xcos^2(x)

解答

\int_{0}^{\frac{π}{2}} 5 e^{x} cos^{2} (x) d x

2 e^{\frac{π}{2}} - 3

十进制

6.62095 \dots

求解步骤

\int_{0}^{\frac{π}{2}} 5 e^{x} cos^{2} (x) d x

提出常数:

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

= 5 \cdot \int_{0}^{\frac{π}{2}} e^{x} cos^{2} (x) d x

使用分布积分法

= 5 [e^{x} cos^{2} (x) - \int - e^{x} sin (2 x) d x]_{0}^{\frac{π}{2}}

\int - e^{x} sin (2 x) d x = \frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x)

= 5 [e^{x} cos^{2} (x) - (\frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x))]_{0}^{\frac{π}{2}}

化简

5 [e^{x} cos^{2} (x) - (\frac{2}{5} e^{x} cos (2 x) - \frac{1}{5} e^{x} sin (2 x))]_{0}^{\frac{π}{2}} : 5 [\frac{1}{5} (5 e^{x} cos^{2} (x) - 2 e^{x} cos (2 x) + e^{x} sin (2 x))]_{0}^{\frac{π}{2}}

= 5 [\frac{1}{5} (5 e^{x} cos^{2} (x) - 2 e^{x} cos (2 x) + e^{x} sin (2 x))]_{0}^{\frac{π}{2}}

带入上下限计算后得:

\frac{2 e ^{\frac{π}{2}}}{5} - \frac{3}{5}

= 5 (\frac{2 e ^{\frac{π}{2}}}{5} - \frac{3}{5})

化简

= 2 e^{\frac{π}{2}} - 3

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