integral from 0 to pi of sin^2(x)e^x

解答

\int_{0}^{π} sin^{2} (x) e^{x} d x

\frac{2 e ^{π} - 2}{5}

十进制

8.85627 \dots

求解步骤

\int_{0}^{π} sin^{2} (x) e^{x} d x

使用分布积分法

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

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

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

化简

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

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

带入上下限计算后得:

\frac{2 e ^{π}}{5} - \frac{2}{5}

= \frac{2 e ^{π}}{5} - \frac{2}{5}

化简

= \frac{2 e ^{π} - 2}{5}

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