Let f(x)+2 f ( 1 x )=x^2+5 and 2 g(x)-3 g ( 1 2 )=x, x 0 . If = ₁^2 f(x) d x , and = ₁^2 g(x) d x , then the value of 9 + is:

Step-by-step solution

aligned & f(x)+2 f ( 1 x )=x^2+5 & f ( 1 x )+2 f(x)= 1 x^2 +5 & f(x)= 2 3 x^2 - x^2 3 + 5 3 & = ₁^2 ( 2 3 x^2 - x^2 3 + 5 3 ) dx & (- 2 3 x - x^3 9 + 5 x 3 )₁^2 & - 1 3 - 8 9 + 10 3 + 2 3 + 1 9 - 5 3 & =2- 7 9 = 11 9 & 2 g(x)-3 g ( 1 2 )=x & ~g ( 1 2 )=- 1 2 & ~g ( x )= x 2 - 3 4 & = ₁^2 ( x 2 - 3 4 ) d x & ( x^2 4 - 3 x 4 )₁^2=1- 3 2 - 1 4 + 3 4 =0 & 9 + =11 & option (4) aligned