Integrals Of Some Particular Functions

Let x = cos

%5E2

%5Ctheta

so that dx = - 2a sin

%5Ctheta

cos

%5Ctheta

%5Ctheta

Now

%5Cint%5E%7B%5Ea%7D_%7B0%7D%5Csqrt%7B%5Cfrac%7Ba-x%7D%7Bx%7D%7Ddx%3D%5Cint%5E%7B0%7D_%7B%5Cfrac%7B%5Cpi%7D%7B2%7D%7D%5Csqrt%7B%5Cfrac%7Ba-a%20cos%5E%7B2%7D%20%5Ctheta%7D%7Ba%20cos%5E%7B2%7D%20%5Ctheta%7D%7D

%5Cquad%5Cquad%5Cquad%5Cquad%5Cquad%5Cleft(-a%20sin%20%5Ctheta%20cos%20%5Ctheta%20%5Cright)d%20%5Ctheta

%5Cquad%5Cquad%20%5Cquad%20%5Cquad%20%5Cquad%20%5Cquad%20%5Cquad%20%5Cquad%20%5Cleft%5B%5Cbecause%20%5Ctheta%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%20at%20x%3D0%3B%5Ctheta%3D0%20at%20x%20%3Da%20%5Cright%5D

%3Da%5Cint%5E%7B%5E%7B%5Cpi%2F2%7D%7D_%7B_0%7D%5Csqrt%7B%5Cfrac%7B1-cos%5E%7B2%7D%20%5Ctheta%7D%7Bcos%5E%7B2%7D%20%5Ctheta%20%7D%7D

2 sin

%5Ctheta

cos

%5Ctheta

%5Ctheta

%5Cquad

%5Cquad

%5Cquad

%5Cquad

%5Cleft%5B%5Cbecause%20%5Cint%5E%7B0%7D_%7Bt%7D%20f%20%5Cleft(x%5Cright)dx%3D-%5Cint%5E%7B0%7D_%7Bt%7D%20f%20%5Cleft(x%5Cright)dx%5Cright%5D

%3Da%5Cint%5E%7B%5Cpi%2F2%7D_%7B0%7D2%5Cfrac%7Bsin%20%5Ctheta%20%7D%7Bcos%20%5Ctheta%20%7D.sin%20%5Ctheta%20cos%20%5Ctheta%20d%5Ctheta

%3Da%5Cint%5E%7B%5Cpi%2F2%7D_%7B0%7D%202sin%5E%7B2%7D%20%5Ctheta%20d%5Ctheta%20%3Da%5Cint%20%5E%7B%5Cpi%2F%202%7D_%7B0%7D%20%5Cleft(1-cos%202%20%5Ctheta%20%5Cright)d%20%5Ctheta

%5Cquad%5Cquad%5Cquad%5Cquad%5Cleft%5B%5Cbecause%20cos%202%5Ctheta%20%3D1-2sin%5E%7B2%7D%20%5Ctheta%20%5Cright%5D

%3Da%5Cleft%5B%5Ctheta%20-%5Cfrac%7Bsin%202%5Ctheta%20%7D%7B2%7D%5Cright%5D%5E%7B%5E%7B%5Cpi%2F2%7D%7D_%7B_%7B_0%7D%7D

%3Da%5Cleft%5B%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D-%5Cfrac%7Bsin%20%5Cpi%7D%7B2%7D%5Cright)-%5Cleft(0-0%5Cright)%5Cright%5D

%3Da%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D-0%5Cright)%3D%5Cfrac%7Ba%5Cpi%7D%7B2%7D

%5CDelta%20(x)%20%3D%20%5Cbegin%7Bvmatrix%7D1%26%5Ccos%20x%261%20-%5Ccos%20x%5C%5C%201%2B%20%5Csin%20x%20%26%20%5Ccos%20x%20%261%2B%20%5Csin%20x%20-%20%5Ccos%20x%5C%5C%20%5Csin%20x%20%26%5Csin%20x%261%5Cend%7Bvmatrix%7D%2C

Applying

C_%7B3%7D%20%5Cto%20C_%7B3%7D%20%2B%20C_%7B2%7D%20-%20C_%7B1%7D

%5CDelta%5Cleft(x%5Cright)%20%3D%20%5Cbegin%7Bvmatrix%7D1%26%5Ccos%20x%20%260%5C%5C%201%20%2B%5Csin%20x%26%5Ccos%20x%20%260%5C%5C%20%5Csin%20x%26%5Csin%20x%20%261%5Cend%7Bvmatrix%7D

%3D%20%5Ccos%20x%20-%20%5Ccos%20x%20%5Cleft(%201%20%2B%20%5Csin%20x%20%5Cright)

%5B%5Cbecause

expanding along

C_%7B3%7D

]

%3D%20-%20%5Ccos%20x.%20%5Csin%20x%20%3D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%20%5C%202x

%5Ctherefore%20%5Cint%5Climits%5E%7B%5Cpi%20%2F4%7D_%7B0%7D%20%5CDelta%5Cleft(x%5Cright)dx

%3D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Cint%5Climits%5E%7B%5Cpi%2F%204%7D_%7B0%7D%20%5Csin%20%5C%202x%20%5C%20dx

%3D%20-%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B-%20%5Cfrac%7B%5Ccos2x%7D%7B2%7D%5Cright%5D%5E%7B%5Cpi%20%2F%204%7D_%7B0%7D

%3D%20%2B%20%5Cfrac%7B1%7D%7B2%20%5Ctimes2%7D%20%5Cleft%5B%20%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20-%20%5Ccos0%5E%7B%5Ccirc%7D%5Cright%5D

%3D%20%5Cfrac%7B1%7D%7B4%7D%20%5Cleft(0-1%5Cright)%20%3D%20-%20%5Cfrac%7B1%7D%7B4%7D

Chapter Questions
8 MCQs