Let (0, 2 ) be fixed. If the integral x+ x- d x=A(x) 2 +B(x) 2 +c , (where c is a constant of integration), then functions A (x) and B (x) are respectively

Step-by-step solution

Let aligned I & = x+ x- d x & = x x + x x - d x & = x + x x - x ~d x & = (x+ ) (x- ) d x aligned Let x- = t aligned I & = ( t +2 ) t & = ( t ) 2 + ( t ) 2 ( t ) d x & = 2 1 dt + 2 ( t ) dt & = 2 t + 2 | ( t )|+ c I & =(x- ) 2 + | (x- )| 2 + c & But x+ x- d x= A (x) 2 + B (x) 2 + c ... [Given] & A (x)=x- , B (x)= | (x- )|+ c aligned