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

解答

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

\frac{3 e ^{\frac{π}{2}} - 2}{5}

十进制

2.48628 \dots

求解步骤

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

使用分布积分法

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

\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}^{\frac{π}{2}}

化简

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

= [\frac{1}{5} (5 e^{x} sin^{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{3 e ^{\frac{π}{2}}}{5} - \frac{2}{5}

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

化简

= \frac{3 e ^{\frac{π}{2}} - 2}{5}

\int_{0}^{2} 3 x 4 - x^{2} d x

\int_{- 11}^{1} x 5 - x d x

\int_{0}^{2} - x^{4} + 2 x^{2} + 8 d x

\int_{5}^{10} \frac{18}{25} x d x

\int_{- 2}^{1} (3 x^{2} + 2) d x