Thanks.

Here is what I got. Expand the e^sinx to infinite series. It is not difficult to calculate int[0, pi/2]sinx^n . Finally, get A*pi + B, where A and B are infinite series
involved 1/(1*3*...*2k-1)^2 and 1/(2*4*...*2k)^2. Since it is a question in high school test, I am wondering if there is a trick to get the exact value of A and B, or something is wrong. The reasonable question to this level is sinx^3.