Some Properties Of Definite Integrals

Let

I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7Bx%20%5Ctan%20x%7D%7B%5Csec%20x%2B%5Ctan%20x%7D%20d%20x

I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7B(%5Cpi-x)%20%5Ctan%20(%5Cpi-x)%7D%7B%5Csec%20(%5Cpi-x)%2B%5Ctan%20(%5Cpi-x)%7D%20d%20x

%5Cleft%5B%5Cbecause%20%5Cint_%7B0%7D%5E%7Ba%7D%20f(x)%20d%20x%3D%5Cint_%7B0%7D%5E%7Ba%7D%20f(a-x)%20d%20x%5Cright%5D

I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7B(%5Cpi-x)%20%5Ctan%20x%7D%7B%5Csec%20x%2B%5Ctan%20x%7D%20d%20x

%5CRightarrow%202%20I%3D%5Cpi%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Cfrac%7B%5Ctan%20x%7D%7B%5Csec%20x%2B%5Ctan%20x%7D%20d%20x

%5CRightarrow%202I%20%3D%5Cpi%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%20%5Ctan%20x(%5Csec%20x-%5Ctan%20x)%20d%20x

%5CRightarrow%202%20I%20%3D%5Cpi%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cleft(%5Csec%20x%20%5Ctan%20x-%5Ctan%20%5E%7B2%7D%20x%5Cright)%20d%20x

%5CRightarrow%202%20I%3D%5Cpi%20%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Cleft(%5Csec%20x%20%5Ctan%20x-%5Csec%20%5E%7B2%7D%20x%2B1%5Cright)%20d%20x

%5CRightarrow%202I%20%3D%5Cpi%5B%5Csec%20x-%5Ctan%20x%2Bx%5D_%7B0%7D%5E%7B%5Cpi%7D

%5CRightarrow%20I%20%3D%5Cpi%5B(%5Csec%20%5Cpi-%5Ctan%20%5Cpi%2B%5Cpi)-(%5Csec%200-%5Ctan%200%2B0)%5D

%5CRightarrow%20I%3D%5Cpi%5Cleft%5B%5Cfrac%7B%5Cpi%7D%7B2%7D-1%5Cright%5D

Chapter Questions
1 MCQs