₂^4 x^2 x^2+ (36-12 x+x^2 ) d x is equal to

Step-by-step solution

Let I = ₂^4 x^2 x^2 + (36-12x+x^2) dx . Observe that 36 - 12x + x^2 = (6-x)^2 , so the integral becomes: I = ₂^4 x^2 x^2 + (6-x)^2 dx Using the substitution property _a^b f(x)dx = _a^b f(a+b-x)dx with a=2 , b=4 : I = ₂^4 (6-x)^2 (6-x)^2 + x^2 dx Adding these two expressions for I : 2I = ₂^4 x^2 + (6-x)^2 x^2 + (6-x)^2 dx = ₂^4 1dx Evaluating the integral yields 2I = [x]₂^4 = 2 , hence I = 1 . Final answer: 1