Step 1: Understanding the Question:
We need to solve for the upper limit \( a \) of a definite integral and then find the sum of all its possible solutions.
Step 2: Key Formula or Approach:
Integrate the function and apply the Fundamental Theorem of Calculus: \( [F(x)]_1^a = 5 \).
Step 3: Detailed Explanation:
\[ \int_1^a (2x + 1) \, dx = 5 \]
Integrating:
\[ [x^2 + x]_1^a = 5 \]
Substitute limits:
\[ (a^2 + a) - (1^2 + 1) = 5 \]
\[ a^2 + a - 2 = 5 \]
\[ a^2 + a - 7 = 0 \]
This is a quadratic equation in \( a \).
The sum of roots (sum of all possible values of \( a \)) is given by \( -B/A \).
Here \( A = 1, B = 1, C = -7 \).
Sum \( = -\frac{1}{1} = -1 \).
Step 4: Final Answer:
The sum of all values of \( a \) is \( -1 \).