Find the area bounded by the y-axis and the curve $x = e^{y} sin π y, y = 0, y = 1$ .

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Solution

$\int_{0}^{1} e^{y} sin (π y) d y$

apply integration by parts: $u = e^{y}, v^{'} = sin (π y)$

$= \frac{e + 1}{π} - \int_{0}^{1} - \frac{1}{π} e^{y} cos (π y) d y$

$= \frac{e + 1}{π} - (- \frac{1}{π} \cdot \int_{0}^{1} e^{y} cos (π y) d y)$

apply integration by parts: $u = e^{y}, v^{'} = cos (π y)$

$= \frac{e + 1}{π} - (- \frac{1}{π} (0 - \int_{0}^{1} e^{y} \frac{1}{π} sin (π y) d y))$

$= \frac{e + 1}{π} - (- \frac{1}{π} (0 - \frac{1}{π} \cdot \int_{0}^{1} e^{y} sin (π y) d y))$

$\int_{0}^{1} e^{y} sin (π y) d y = \frac{e + 1}{π} - (- \frac{1}{π} (0 - \frac{1}{π} \cdot \int_{0}^{1} e^{y} sin (π y) d y))$

$= \frac{π e + π}{π^{2} + 1}$

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