By using the properties of definite integrals, evaluate the integrals in \int\limits_{\begin{smallmatrix}   0\\\end{smallmatrix}}^ {\pi }{{}} log(1+cosx)dx
By using the properties of definite integrals, evaluate the integrals in \int\limits_{\begin{smallmatrix}   0\\\end{smallmatrix}}^ {\pi }{{}} log(1+cosx)dx

Let I =         ……….(i)

 I = 

    ……….(ii)

Adding eq. (i) and (ii),

2I = 

 2I = 

 I =  ……….(iii)

 I = 

     ……….(iv)

Adding eq. (i) and (ii),

2I = 

 I = 

 I = 

 I = I1 –  ……….(v)

Where I1 =  …….(vi)

Putting  in eq. (vi),

 

 

Limits of integration when  and 

 From eq. (vi),

I1 = 

 I1 = 

 I1 =  [From eq. (iii)]

Putting this value in eq. (v), I = 

 2I = 

 2I – I = 

 I =