If for a continuous function f ( x ) , _ - ^t(f(x)+x d x)= ^2-t^2 , for all t - , then f (- 3 ) is equal to:

Step-by-step solution

Let _ - ^t(f(x)+x) d x= ^2-t^2 aligned & _ - ^t f(x) d x+ _ - ^t x d x= ^2-t^2 & _ - ^t f(x) d x+ ( t^2 2 - ^2 2 )= ^2-t^2 & _ - ^t f(x) d x= 3 2 ( ^2-t^2 ) aligned differentiating with respect to t d d t [ _ - ^t f(x) d x ]= 3 2 d d t ( ^2-t^2 )f(t) d t d t -f(- ) d d t (- )=-3 tf(t)=-3 tf (- 3 )=-3 (- 3 )=