Solution 1: If $x=1$, then $3x+1=4$, which is an even integer. In this case, the five given choices are (A) $x+3=4$, (B) $x-3=-2$, (C) $2x=2$, (D) $7x+4=11$, (E) $5x+3=8$. Of these, the only odd integer is (D). Therefore, since $x=1$ satisfies the initial criteria, then (D) must be the correct answer as the result must be true no matter what integer value of $x$ is chosen that makes $3x+1$ even. Solution 2: If $x$ is an integer for which $3x+1$ is even, then $3x$ is odd, since it is 1 less than an even integer. If $3x$ is odd, then $x$ must be odd (since if $x$ is even, then $3x$ would be even). If $x$ is odd, then $x+3$ is even (odd plus odd equals even), so (A) cannot be correct. If $x$ is odd, then $x-3$ is even (odd minus odd equals even), so (B) cannot be correct. If $x$ is odd, then $2x$ is even (even times odd equals even), so (C) cannot be correct. If $x$ is odd, then $7x$ is odd (odd times odd equals odd) and so $7x+4$ is odd (odd plus even equals odd). If $x$ is odd, then $5x$ is odd (odd times odd equals odd) and so $5x+3$ is even (odd plus odd equals even), so (E) cannot be correct. Therefore, the one expression which must be odd is $7x+4$.