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

Question

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

Answer 1

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.