Definite Integration Question - None

Question

Suppose that a continuous function $f(x)$ satisfies the relation $\int_{x}^{x+1} f(t)\, dt = e^x$ for every $x \geq 0$. The value of $f(2) - f(0)$, equals:

Accepted Answer

Step 1: Differentiate both sides of $\int_{x}^{x+1} f(t)\,dt = e^x$ with respect to $x$ using Leibniz rule:

$$\frac{d}{dx}\int_{x}^{x+1} f(t)\,dt = \frac{d}{dx}(e^x)$$

Step 2: Apply Leibniz rule: $\frac{d}{dx}\int_{a(x)}^{b(x)} f(t)\,dt = f(b(x))\cdot b'(x) - f(a(x))\cdot a'(x)$

$$f(x+1)\cdot(1) - f(x)\cdot(1) = e^x$$

$$f(x+1) - f(x) = e^x$$

Step 3: Use this relation for specific values. For $x = 0$: $f(1) - f(0) = e^0 = 1$

For $x = 1$: $f(2) - f(1) = e^1 = e$

Step 4: Add these equations:

$$[f(1) - f(0)] + [f(2) - f(1)] = 1 + e$$

$$f(2) - f(0) = 1 + e$$

∴ Answer: B

Question:

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