Question

Let a differentiable function $f$ satisfy $f(x)+\int\limits_3^x \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3$ Then $12 f(8)$ is equal to :

• A. 19
• B. 1
• C. 17
• D. 34

Hint:

When differentiating an equation involving an integral, use the Fundamental Theorem of Calculus to simplify and solve the problem efficiently.

Answer 1

The Correct Option is C

Step 1: Start with the given equation: \[ f(x) + \int_3^x f(t) \, dt = \sqrt{x+1}. \] Step 2: Differentiate both sides with respect to \(x\): \[ \frac{d}{dx} \left( f(x) + \int_3^x f(t) \, dt \right) = \frac{d}{dx} \left( \sqrt{x+1} \right) \] Using the Fundamental Theorem of Calculus, this becomes: \[ f'(x) + f(x) = \frac{1}{2\sqrt{x+1}}. \] Step 3: Use the method of integrating factors (I.F.): The integrating factor is \(e^x\), so multiply both sides by \(e^x\): \[ e^x f'(x) + e^x f(x) = \frac{e^x}{2\sqrt{x+1}}. \] Step 4: Now integrate both sides: \[ \int e^x f'(x) \, dx + \int e^x f(x) \, dx = \int \frac{e^x}{2\sqrt{x+1}} \, dx. \] The left side simplifies to \(e^x f(x)\). Now, perform the integration on the right-hand side. After integration and substitution, you'll arrive at: \[ f(x) = \frac{(x + 1)^{3/2}}{3\sqrt{x+1}} + C. \] Step 5: Use the condition that \(f(3) = 2\) to solve for \(C\): \[ f(3) = 2 \quad \Rightarrow \quad C = \frac{16}{3}. \] Step 6: Calculate \(f(8)\): \[ f(8) = \frac{34}{24} \quad \Rightarrow \quad 12f(8) = 17. \]