The value of ₀^ (1+ x) d x is

Step-by-step solution

Let aligned I &= ₀^ (1+ x) d x...(i) I &= ₀^ 1+ ( -x) d x &= ₀^ (1- x) d x...(ii) aligned On adding Eqs. (i) and (ii), we get aligned 2 I &= ₀^ (1+ x)+ (1- x) d x I &= 1 2 ₀^ (1- ² x ) d x &= 1 2 ₀^ ² x d x &= ₀^ x d x &=2 ₀^ / 2 x d x aligned aligned ₀^ 2 a f(x) d x=2 ₀^ a f(x) d x, . & if f(2 a-x)=f(x) =2 - 2 2 ( ₀^ / 2 x d x=- 2 2 ) = 1 2 aligned