Let f:[1, ) [2, ) be a differentiable function, If 10 ₁^ x f( t ) dt =5 x f( x )- x ^5-9 for all x 1 , then the value of f(3) is :

Step-by-step solution

aligned & 10 ~d dx ₁^ x f ( t ) dt = d dx (5 xf ( x )- x ^5-9 ) & 10 f(x)=5 f(x)+5 xf ^ ( x )-5 x ^4 & f ( x )+ x ^4= xf ^ ( x ) & y + x ^4= x dy dx & d y d x +y (- 1 x )=x^3 & y e^ - 1 x d x = x^3 e^ - 1 x d x +c & ye ^ - tn |x| = x ^3 e ^ - tr |x| + c & y | x | = x ^3 | x | + c & y x = x ^2+ c & y x = x ^3 3 + c aligned Put x=1 in given equation aligned & 0=5 f (1)-1-9 f (1)=2 & 2 1 = 1 3 + c c = 5 3 & f (3) 3 = 27 3 + 5 3 & f (3)=32 aligned