Please comment this homework of mine

[sandra1]

Homework Statement

Give a counter example to negate the statement that for functions f_1, f_2 :[a, infinity) --> R with c_1 and c_2 are real constant, we have:

∫[a,∞] (c_1)(f_1) +(c_2)(f_2) dx = c_1 ∫[a,∞]f_1 dx + c_2 ∫[a,∞] f_2 dx

Homework Equations

The Attempt at a Solution

Choose f_1(x) = x
f_2(x) = -x
c_1 = c_2 = 2
then ∫[a,∞] (c_1)(f_1) dx = lim [b->∞] ∫[a,b] 2x + (-2x) dx
= lim [b->∞] ∫[a,b] 0 dx = lim [b->∞] C = C (with some constant C)

while:
c_1 ∫[a,∞]f_1 dx + c_2 ∫[a,∞] f_2 dx =
2 lim[b_1->∞]∫[a,b_1]x dx + 2 lim[b_2->∞]∫[a,b_2](-x) dx
= 2 lim[b_1->∞] (1/2)(x^2) [a,b_1] + 2 lim[b_2->∞] (-1/2)(x^2) [a,b_2]
= lim[b_1->∞] (b_1^2) - (a)^2 + lim[b_2->∞] -(b_2)^2 + (a)^2
= lim[b_1->∞] (b_1^2) + lim[b_2->∞] -(b_2)^2 = ∞ - ∞

 

[kurt.physics]

= undefinedHence, for this counter example, we have: ∫[a,∞] (c_1)(f_1) +(c_2)(f_2) dx ≠ c_1 ∫[a,∞]f_1 dx + c_2 ∫[a,∞] f_2 dx