If f (5-x)= f (x) and ₂^3 f (x) d x=2 then ₂^3 x f (x) d x=

Step-by-step solution

Given the definite integral I = ₂^3 x f(x) dx with the conditions f(5-x)=f(x) and ₂^3 f(x) dx=2 , apply the substitution property _a^b g(x) dx = _a^b g(a+b-x) dx . Since a=2 and b=3 , then a+b=5 and I = ₂^3 (5-x) f(5-x) dx . Using the given symmetry f(5-x)=f(x) , this becomes I = ₂^3 (5-x) f(x) dx . Expanding the integrand: I = 5 ₂^3 f(x) dx - ₂^3 x f(x) dx . The first term is 5 2 = 10 by the given integral value, and the second term is I . This yields I = 10 - I , so 2I = 10 and I = 5 . Final answer: 5