Integral

Given,

I_%7Bn%7D%20%3D%5Cint%20%5Cfrac%7B%5Csin%20n%20x%7D%7B%5Csin%20x%7D%20d%20x%20%5Cdots

(i)

I_%7Bn-2%7D%20%3D%5Cint%20%5Cfrac%7B%5Csin%20(n-2)%20x%7D%7B%5Csin%20x%7D%20d%20x%20%5Cdots

(ii) Subtracting E (ii) from E (i), we get

I_%7Bn%7D-I_%7Bn-2%7D%3D%5Cint%20%5Cfrac%7B%5C%7B%5Csin%20n%20x-%5Csin%20(n-2)%20x%5C%7D%7D%7B%5Csin%20x%7D%20d%20x

%3D%5Cint%20%5Cfrac%7B2%20%5Ccos%20(n-1)%20x%20%5Csin%20x%7D%7B%5Csin%20x%7D%20d%20x%3D%5Cint%202%20%5Ccos%20(n-1)%20x%20d%20x

%3D%5Cfrac%7B2%20%5Csin%20(n-1)%20x%7D%7B(n-1)%7D

%5Ctherefore%20I_%7B6%7D-I_%7B4%7D%3D%5Cfrac%7B2%20%5Csin%205%20x%7D%7B5%7D

and

I_%7B4%7D-I_%7B2%7D%3D%5Cfrac%7B2%20%5Csin%203%20x%7D%7B3%7D

Now,

I_%7B2%7D%3D%5Cint%20%5Cfrac%7B%5Csin%202%20x%7D%7B%5Csin%20x%7D%20d%20x

%3D%5Cint%20%5Cfrac%7B2%20%5Csin%20x%20%5Ccos%20x%7D%7B%5Csin%20x%7D%20d%20x%3D2%20%5Cint%20%5Ccos%20x%20d%20x

%3D2%20%5Csin%20x%2Bc

and

I_%7B6%7D%3DI_%7B4%7D%2B2%20%5Cfrac%7B%5Csin%203%20x%7D%7B5%7D

%3DI_%7B2%7D%2B2%20%5Cfrac%7B%5Csin%203%20x%7D%7B3%7D%2B2%20%5Cfrac%7B%5Csin%205%20x%7D%7B5%7D

%3D2%20%5Cfrac%7B%5Csin%205%20x%7D%7B5%7D%2B2%20%5Cfrac%7B%5Csin%203%20x%7D%7B3%7D%2B2%20%5Csin%20x%2Bc

%3D%5Cfrac%7B2%20%5Csin%205%20x%7D%7B5%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Cleft(3%20%5Csin%20x-4%20%5Csin%20%5E%7B3%7D%20x%5Cright)%2B2%20%5Csin%20x%2Bc

I_%7B6%7D%3D%5Cfrac%7B2%7D%7B5%7D%20%5Csin%205%20x-%5Cfrac%7B8%7D%7B3%7D%20%5Csin%20%5E%7B3%7D%20x%2B4%20%5Csin%20x%2Bc

Given integral

I(x)%3D%5Cint%20x%5E%7B2%7D(%5Clog%20x)%5E%7B2%7D%20d%20x%3D%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D(%5Clog%20x)%5E%7B2%7D-%5Cint%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%20%5Cfrac%7B2%20%5Clog%20x%7D%7Bx%7D%20d%20x

[by integration by parts]

%3D%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D(%5Clog%20x)%5E%7B2%7D-%5Cfrac%7B2%7D%7B3%7D%5Cleft%5B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D(%5Clog%20x)-%5Cint%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%20d%20x%5Cright%5D

%3D%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D(%5Clog%20x)%5E%7B2%7D-%5Cfrac%7B2%7D%7B3%7D%5Cleft%5B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D(%5Clog%20x)-%5Cfrac%7B1%7D%7B3%7D%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%5Cright%5D%2BC

%3D%5Cfrac%7Bx%5E%7B3%7D%7D%7B27%7D%5Cleft%5B9(%5Clog%20x)%5E%7B2%7D-6(%5Clog%20x)%2B2%5Cright%5D%2BC

%5Cbecause%20I(1)%3D0

%5Ctherefore%20%5Cfrac%7B2%7D%7B27%7D%2BC%3D0

%5CRightarrow%20C%3D-%5Cfrac%7B2%7D%7B27%7D

%5Ctherefore%20I(x)%3D%5Cfrac%7Bx%5E%7B3%7D%7D%7B27%7D%5Cleft%5B9(%5Clog%20x)%5E%7B2%7D-6(%5Clog%20x)%2B2%5Cright%5D-%5Cfrac%7B2%7D%7B27%7D

Chapter Questions
2 MCQs