Loading [MathJax]/jax/output/CommonHTML/jax.js

Integral exercise solved using Property of u-Substitution baf(x)dx=baf(a+bx)dx


Exercise:

Compute, 2π0xsin2nxsin2nx+cos2nxdx  where nN

Solution: Let I=2π0xsin2nxsin2nx+cos2nxdx

Using the property a0f(x)dx=a0f(ax)dx to get:

I=2π0xsin2nxsin2nx+cos2nxdx=2π0(2πx)sin2n(2πx)sin2n(2πx)+cos2n(2πx)dx

   =2π02πsin2n(2πx)sin2n(2πx)+cos2n(2πx)2π0xsin2n(2πx)sin2n(2πx)+cos2n(2πx)dx

   =2π2π0sin2n(2πx)sin2n(2πx)+cos2n(2πx)dxI

2I=2π2π0sin2n(2πx)sin2n(2πx)+cos2n(2πx)dx

I=π2π0sin2n(2πx)sin2n(2πx)+cos2n(2πx)dx

      =π2π0sin2n(2πx)sin2n(2πx)+cos2n(2πx)dx=π2π0(sinx)2n(sinx)2n+(cosx)2ndx

       =π2π0sin2nxsin2nx+cos2nxdx

But =2π0sin2nxsin2nx+cos2nxdx=π20sin2nxsin2nx+cos2nxdx+2ππ2sin2nxsin2nx+cos2nxdx

But I1=π20sin2nxsin2nx+cos2nxdx=π20sin2n(π2x)sin2n(π2x)+cos2n(π2x)dx=π20cos2nxcos2nx+sin2nxdx

So 2I1=π20sin2nx+cos2nxsin2nx+cos2nxdx=π2I1=π4

Also I2=2ππ2sin2nxsin2nx+cos2nxdx=2ππ2sin2n(5π2x)sin2n(5π2x)+cos2n(5π2x)dx

=2ππ2sin2n(π2x)sin2n(π2x)+cos2n(π2x)dx=2ππ2cos2nxcos2nx+sin2nxdx

So 2I2=2ππ2sin2nx+cos2nxsin2nx+cos2nxdx=2ππ2=3π2I2=3π4

Hence 2π0sin2nxsin2nx+cos2nxdx=I1+I2=π4+3π4=π

Therefore 2π0xsin2nxcos2nx+sin2nxdx=π2

No comments:

Post a Comment