If f(x) = ₁^x t (1+t)^3 dt , then the value of f(3) + f ( 1 3 ) is equal to

Step-by-step solution

f ( 1 3 ) = ₁^ 1/3 t (1+t)^3 dt Let t = 1 u dt = - 1 u^2 du f ( 1 3 ) = ₁^3 - u (1+ 1 u )^3 (- 1 u^2 ) du = ₁^3 u u (1+u)^3 du Now, f(3) + f ( 1 3 ) = ₁^3 t (1+t)^3 dt + ₁^3 t t (1+t)^3 dt = ₁^3 (1+t) t (1+t)^3 dt = ₁^3 t (1+t)^2 dt Using integration by parts, let u = t and dv = 1 (1+t)^2 dt t (1+t)^2 dt = - t 1+t - (- 1 1+t ) 1 t dt = - t 1+t + ( 1 t - 1 1+t ) dt = - t 1+t + t - (1+t) = t t 1+t - (1+t) Evaluating from 1 to 3 : = ( 3 3 4 - 4 ) - ( 0 - 2 ) = 3 4 3 - 2 2 + 2 = 3 4 3 - 2 . Answer: 3 4 3 - 2