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I have tried to solve $\int _{ 0 }^{ \frac { \pi }{ 2 } } \int _{ 0 }^{ x }{ { e }^{ sin(y) } } { sin(x)dydx }$ . I have solved it by using mathematica to evaluate $\int { { e }^{ sin(x) }dx } $ but it is turning tedious so hoping to get some insight into solve it in a better way

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Hint Changing the order of integration you get $$\int _{ 0 }^{ \frac { \pi }{ 2 } } \int _{ 0 }^{ x }{ { e }^{ \sin(y) } } { \sin(x)dydx }=\int _{ 0 }^{ \frac { \pi }{ 2 } } \int _{ y }^{ \frac { \pi }{ 2 } }{ { e }^{ \sin(y) } } { \sin(x)dxdy }$$ which is easy to integrate.

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  • $\begingroup$ what is the principle behind it? $\endgroup$ – SHOURIE MRSS Jan 12 at 4:04
  • $\begingroup$ I got the expected answer e-1, but I am still not able to spot the rationale behind the suggestion? $\endgroup$ – SHOURIE MRSS Jan 12 at 4:20
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    $\begingroup$ @SHOURIEMRSS Whenever when you need to integrate a double integral $$\int \int f(x,y) dy dx$$ and your function is hard/impossible to integrate with respect to $y$ but easy to integrate with respect to $x$ it is natural to change the order of integration.... Start with the easy step, and see what you get, maybe the second integral is also easy....... This is a pretty standard exercise of order of integration. $\endgroup$ – N. S. Jan 12 at 4:30

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