Integrals Of Some Particular Functions

Given,

f%20%5Cleft(0%5Cright)%3D9

and

f%20%5Cleft(x%5Cright)%3D%5Cfrac%7Bsin%5Cleft(%5Cfrac%7B9x%7D%7B2%7D%5Cright)%7D%7Bsin%5Cfrac%7Bx%7D%7B2%7D%7D

Now,

f%20%5Cleft(-x%5Cright)%3D%5Cfrac%7Bsin%5Cleft(-%5Cfrac%7B9x%7D%7B2%7D%5Cright)%7D%7Bsin%20%5Cleft(-%5Cfrac%7Bx%7D%7B2%7D%5Cright)%7D%20%3D%5Cfrac%7B-sin%20%5Cleft(%5Cfrac%7B9x%7D%7B2%7D%5Cright)%7D%7B-sin%20%5Cleft(%5Cfrac%7Bx%7D%7B2%7D%5Cright)%7D%3Df%20%5Cleft(x%5Cright)

Hence,

f%5Cleft(x%5Cright)

is an even function

%5Ctherefore%20I%3D%5Cfrac%7B2%7D%7B%5Cpi%7D%20%5Cint_%7B-%5Cpi%7D%5E%7B%5Cpi%7Df%20%5Cleft(x%5Cright)dx

%3D%5Cfrac%7B2%7D%7B%5Cpi%7D%5Ctimes2%20%5Cint%5E%7B%5Cpi%7D_%7B0%7D%20f%20%5Cleft(x%5Cright)dx

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7Bsin%20%5Cfrac%7B9x%7D%7B2%7D%7D%7Bsin%20%5Cfrac%7Bx%7D%7B2%7D%7Ddx

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7Bsin%5Cleft(4x%2B%5Cfrac%7Bx%7D%7B2%7D%5Cright)%7D%7Bsin%20%5Cfrac%7Bx%7D%7B2%7D%7D%20dx

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7Bsin%5C%2C4x%5C%2Ccos%20%5Cfrac%7Bx%7D%7B2%7D%2Bsin%20%5Cfrac%7Bx%7D%7B2%7D%20cos%20%5C%2C4x%7D%7Bsin%20%5Cfrac%7Bx%7D%7B2%7D%7Ddx

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cleft(sin%5C%2C4x%5C%2Ccot%20%5Cfrac%7Bx%7D%7B2%7D%20dx%5Cright)%5Cright%5D%2B0

%5Cleft%5B%5Cbegin%7Bmatrix%7D%5Cbecause%20%5Cint_%7B0%7D%5E%7B%5Cpi%7Dcos%5C%2C4x%5C%2Cdx%3D%5Cleft%5B%5Cfrac%7Bsin%5C%2C4x%7D%7B4%7D%5Cright%5D_%7B0%7D%5E%7B%5Cpi%7D%5C%5C%20%3D%5Cfrac%7B1%7D%7B4%7D%5Cleft(sin%5C%2C4%5Cpi-sin%5Cright)%3D%5Cfrac%7B1%7D%7B4%7D%5Cleft%5B0-0%5Cright%5D%3D0%5Cend%7Bmatrix%7D%5Cright%5D%3D-%5Cfrac%7B1%7D%7B4%7D%5Cleft%5B0-0%5Cright%5D%3D0

%5Ctherefore%20I%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20sin%5C%2C4x%5C%2Ccot%20%5Cfrac%7Bx%7D%7B2%7D%20dx%20%5Cldots%5Cleft(i%5Cright)

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20sin%20%5Cleft(4%5Cpi-4x%5Cright)cot%20%5Cleft(%5Cfrac%7B%5Cpi-x%7D%7B2%7D%5Cright)dx

%5CRightarrow%20I%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20-sin%20%5C%2C4x%20%5C%2C%20tan%20%5Cfrac%7Bx%7D%7B2%7D%20dx%20%5Cldots%5Cleft(ii%5Cright)

On adding Eqs.

%5Cleft(i%5Cright)

and

%5Cleft(ii%5Cright)

, we get

2I%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%20%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20sin%5C%2C4x%5Cleft(cot%20%5Cfrac%7Bx%7D%7B2%7D-tan%5Cfrac%7Bx%7D%7B2%7D%5Cright)dx%5Cright%5D

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7Dsin%5C%2C4x%20%5Cfrac%7B%5Cleft(cos%5E%7B2%7D%5Cfrac%7Bx%7D%7B2%7D-sin%5E%7B2%7D%5Cfrac%7Bx%7D%7B2%7D%5Cright)%7D%7Bsin%20%5Cfrac%7Bx%7D%7B2%7Dcos%20%5Cfrac%7Bx%7D%7B2%7D%7D%20dx%5Cright%5D

%3D%5Cfrac%7B4%7D%7B%5Cpi%7D%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cfrac%7Bsin%5C%2C4x%5C%2Ccos%5C%2Cx%7D%7B%5Cfrac%7B1%7D%7B2%7Dsin%5C%2Cx%7Ddx%5Cright%5D

%3D%5Cfrac%7B8%7D%7B%5Cpi%7D%20%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7B2%5C%2Csin%5C%2C2x%5C%2Ccos%5C%2C2x%5C%2Ccos%5C%2Cx%7D%7Bsin%5C%2Cx%7Ddx%5Cright%5D

%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cfrac%7B2%5C%2Csin%5C%2Cx%5C%2Ccos%5C%2Cx%5C%2Ccos%5C%2C2x%5C%2Ccos%5C%2Cx%7D%7Bsin%5C%2Cx%7Ddx%5Cright%5D

%3D%5Cfrac%7B32%7D%7B%5Cpi%7D%20%5Cint_%7B0%7D%5E%7B%5Cpi%7Dcos%5E%7B2%7D%5C%2Cx%5C%2C%20cos%5C%2C2x%5C%2Cdx

%3D%5Cfrac%7B32%7D%7B%5Cpi%7D%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cleft(%5Cfrac%7Bcos%5C%2C2x%2B1%7D%7B2%7D%5Cright)cos%5C%2C2x%5C%2Cdx

%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7Dcos%5E%7B2%7D%5C%2C2x%5C%2Cdx%2B%5Cint_%7B0%7D%5E%7B%5Cpi%7Dcos%5C%2C2x%5C%2Cdx%5Cright%5D

%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%20%5Cleft%5B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7Bcos%5C%2C4x%2B1%7D%7B2%7Ddx%2B%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20cos%5C%2C2x%5C%2Cdx%5Cright%5D

%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cfrac%7Bsin%5C%2C4x%7D%7B4%7D%2Bx%5Cright%5D_%7B0%7D%5E%7Bx%7D%2B%5Cleft%5B%5Cfrac%7Bsin%5C%2C2x%7D%7B2%7D%5Cright%5D_%7B0%7D%5E%7Bx%7D%5Cright%5D

%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B0%2B%5Cpi-0-0%5Cright%5D%2B%5Cleft%5B%5Cfrac%7B0-0%7D%7B2%7D%5Cright%5D%5Cright%5D%3D%5Cfrac%7B16%7D%7B%5Cpi%7D%5Cleft%5B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D

2I%20%5CRightarrow%208

%5CRightarrow%20I%3D4

Chapter Questions
10 MCQs