integral from 0 to 2/3 of integral from y to 2-2y of (x+2y)e^{(y-x)}

解答

\int_{0}^{\frac{2}{3}} \int_{y}^{2 - 2 y} (x + 2 y) e^{(y - x)} d x d y

\frac{1}{3} + \frac{1}{e ^{2}}

十进制

0.46866 \dots

求解步骤

\int_{0}^{\frac{2}{3}} \int_{y}^{2 - 2 y} (x + 2 y) e^{(y - x)} d x d y

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

= \int_{0}^{\frac{2}{3}} (3 y - 3 e^{3 y - 2} + 1) d y

\int_{0}^{\frac{2}{3}} (3 y - 3 e^{3 y - 2} + 1) d y = \frac{1}{3} + \frac{1}{e ^{2}}

= \frac{1}{3} + \frac{1}{e ^{2}}

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