Exercise:
Compute, ∫2π0xsin2nxsin2nx+cos2nxdx where n∈N∗
Solution: Let I=∫2π0xsin2nxsin2nx+cos2nxdx
Using the property ∫a0f(x)dx=∫a0f(a−x)dx to get:
I=∫2π0xsin2nxsin2nx+cos2nxdx=∫2π0(2π−x)sin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)dx
=∫2π02πsin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)−∫2π0xsin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)dx
=2π∫2π0sin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)dx−I
⇔2I=2π∫2π0sin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)dx
⇔I=π∫2π0sin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)dx
=π∫2π0sin2n(2π−x)sin2n(2π−x)+cos2n(2π−x)dx=π∫2π0(−sinx)2n(−sinx)2n+(cosx)2ndx
=π∫2π0sin2nxsin2nx+cos2nxdx
But =∫2π0sin2nxsin2nx+cos2nxdx=∫π20sin2nxsin2nx+cos2nxdx+∫2ππ2sin2nxsin2nx+cos2nxdx
But I1=∫π20sin2nxsin2nx+cos2nxdx=∫π20sin2n(π2−x)sin2n(π2−x)+cos2n(π2−x)dx=∫π20cos2nxcos2nx+sin2nxdx
So 2I1=∫π20sin2nx+cos2nxsin2nx+cos2nxdx=π2⇔I1=π4
Also I2=∫2ππ2sin2nxsin2nx+cos2nxdx=∫2ππ2sin2n(5π2−x)sin2n(5π2−x)+cos2n(5π2−x)dx
=∫2ππ2sin2n(π2−x)sin2n(π2−x)+cos2n(π2−x)dx=∫2ππ2cos2nxcos2nx+sin2nxdx
So 2I2=∫2ππ2sin2nx+cos2nxsin2nx+cos2nxdx=2π−π2=3π2⇔I2=3π4
Hence ∫2π0sin2nxsin2nx+cos2nxdx=I1+I2=π4+3π4=π
Therefore ∫2π0xsin2nxcos2nx+sin2nxdx=π2
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