Let f(x) = xsinx, put -x instead of x then we get f(−x)=(−x)sin(−x)=xsinx . Therefore, f(-x) = f(x) implies that f(x) is an even function. Thus, the integral is given by,
π
2
∫
−π
2
xsinxdx=
π
2
∫
0
xsinxdx =2[x(−cosx)−∫(−cosxdx)]
π
2
=2[−xcosx+sinx]
π
θ
=2[(−
π
2
cos(
π
2
)+sin(
π
2
))−(−0cos0+sin0)] =2 Hence, the value of the given integral is 2.