The value of the definite integral ₀^ / 2 ^2 t ( t) d t can be expressed as ^a b (c- d) , where a, b, c and d are integers. Find the smallest possible value of a+b+c+d .

Step-by-step solution

I= ₀^ / 2 ^2 t ( t) d t= ₀^ / 2 1 2 (1- 2 t) ( t) d t .= 1 2 (t- 2 t 2 ) ( t) ]₀^ / 2 - 1 2 ₀^ / 2 t t d t+ 1 4 ₀^ / 2 2 t t d t (Using I.B.P.) .=0- 1 2 t t ]₀^ / 2 + 1 2 ₀^ / 2 t d t+ 1 2 ₀^ / 2 ^2 t d t =0-0- 4 2+ 1 4 ₀^ / 2 (1+ 2 t) d t [Note: . ₀^ / 2 ( x) d x= - 2 2 ] =- 4 2+ 1 4 [t+ 2 t 2 ]₀^ / 2 =- 4 2+ 8 = 8 (1- 4) Therefore, a+b+c+d=1+8+1+4=14