# Need help in solving $\int _{ 0 }^{ \frac { \pi }{ 2 } } \int _{ 0 }^{ x }{ { e }^{ sin(y) } } { sin(x)dydx }$

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

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.
• @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. – N. S. Jan 12 at 4:30