Question

Given \(\int_1^a (2x + 1) \, dx = 5\), find the sum of all values of \(a\).

Hint:

When a question asks for the "sum of all values" of a variable resulting from a quadratic equation, do not waste time finding individual roots (using the discriminant). Use the sum of roots formula \(-B/A\) directly!

Answer 1

Step 1: Understanding the Concept:
This is a definite integral problem where the result of the integration is given as a constant. We need to evaluate the integral in terms of the upper limit \(a\) and solve the resulting quadratic equation.

Step 2: Key Formula or Approach:
1. Power rule of integration: \(\int x^n \, dx = \frac{x^{n+1}}{n+1}\).
2. Fundamental Theorem of Calculus: \(\int_p^q f(x) \, dx = [F(x)]_p^q = F(q) - F(p)\).
3. Sum of roots of a quadratic equation \(Ax^2 + Bx + C = 0\) is \(-B/A\).

Step 3: Detailed Explanation:
First, evaluate the integral:
\[ \int_1^a (2x + 1) \, dx = [x^2 + x]_1^a \] Applying the limits:
\[ (a^2 + a) - (1^2 + 1) = a^2 + a - 2 \] Given that the integral equals 5:
\[ a^2 + a - 2 = 5 \] \[ a^2 + a - 7 = 0 \] This is a quadratic equation in \(a\). We need to find the sum of all possible values of \(a\).
By comparing with \(Ax^2 + Bx + C = 0\), we have \(A = 1\) and \(B = 1\).
Sum of roots (values of \(a\)) \(= -\frac{B}{A} = -\frac{1}{1} = -1\).

Step 4: Final Answer:
The sum of all values of \(a\) is \(-1\).