integral from 0 to pi/4 of e^{-x}sin(x)

解答

\int_{0}^{\frac{π}{4}} e^{- x} sin (x) d x

- \frac{1}{2 e ^{\frac{π}{4}}} + \frac{1}{2}

十进制

0.17760 \dots

求解步骤

\int_{0}^{\frac{π}{4}} e^{- x} sin (x) d x

计算不定积分:

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

带入上下限计算后得:

\int_{0}^{\frac{π}{4}} e^{- x} sin (x) d x = - \frac{1}{2 e ^{\frac{π}{4}}} - (- \frac{1}{2})

= - \frac{1}{2 e ^{\frac{π}{4}}} - (- \frac{1}{2})

化简

= - \frac{1}{2 e ^{\frac{π}{4}}} + \frac{1}{2}

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