The value of ∫−π/2π/2 xsin⁡xex+1dx is equal to

I=∫−π/2π/2 xsin⁡xex+1dx=∫−π/2π/2 (−x)sin⁡(−x)e−x+1dx (Prop. 11) =∫−π/2π/2 (xsin⁡x)exex+1dx   2I=∫−π/2π/2 xsin⁡xex+1+∫−π/2π/2 ex(xsin⁡x)ex+1dx   =∫−π/2π/2 ex+1(xsin⁡x)ex+1dx   =∫−π/2π/2 xsin⁡xdx=2∫0π/2 xsin⁡xdx   ( Prop. 12)     =2−xcos⁡x0π/2+∫0π/2 cos⁡xdx     =2sin⁡x0π/2=2⇒   I=1.