Question

If \( \int_{0}^{x} f(t)dt = x^2 + e^x (x > 0) \), then \( f(1) \) is equal to:

• A. \( 1 + e \)
• B. \( 2 + e \)
• C. \( 3 + e \)
• D. \( e \)
• E. \( 0 \)

Hint:

The Fundamental Theorem of Calculus acts as the bridge between differentiation and integration. Whenever you see a function trapped inside an integral with a variable limit, your first instinct should be to differentiate both sides.

Answer 1

The Correct Option is B

Concept: To find the function \( f(x) \) when given its definite integral from a constant to \( x \), we use the Fundamental Theorem of Calculus (Part 1). It states that: \[ \frac{d}{dx} \int_{a}^{x} f(t)dt = f(x) \] Applying this theorem allows us to differentiate the right-hand side of the equation to find the function itself.

Step 1: Differentiate both sides of the given equation with respect to \( x \).
Given: \[ \int_{0}^{x} f(t)dt = x^2 + e^x \] Differentiating: \[ \frac{d}{dx} \left[ \int_{0}^{x} f(t)dt \right] = \frac{d}{dx} [x^2 + e^x] \] Using the power rule and the exponential derivative rule: \[ f(x) = 2x + e^x \]

Step 2: Evaluate the function at \( x = 1 \).
\[ f(1) = 2(1) + e^1 \] \[ f(1) = 2 + e \]