Let a differentiable function f satisfy f(x) + ∫₃ˣ f(t)/t dt = √(x+1), x ≥ 3. Then 12f(8) is equal t

Q: Why do we differentiate the whole equation? Doesn't that change the function?

Great question! We aren't changing the function; we are transforming the equation into a differential form. Since the original equation holds for all $x \ge 3$, the derivative of the left side must equal the derivative of the right side. It's a standard technique to 'unlock' functions hidden under integral signs.

Q: How did we know to pick $x=3$ to find the constant $C$?

That is the 'magic' of the integral! The integral $\int_{3}^{3} \dots dt$ is always zero, regardless of the integrand. By plugging in the lower limit of the integral, we effectively eliminate the integral term, allowing us to isolate $f(3)$ and find the initial condition needed to solve for $C$.

Q: Is it always a Linear Differential Equation?

In JEE Advanced, if you see a term like $f'(x) + P(x)f(x) = Q(x)$, it is almost certainly a Linear Differential Equation. Always look for that structure. Once you identify $P(x)$ and $Q(x)$, the path is paved with the Integrating Factor method. It's a reliable, powerful tool in your arsenal.

Question

Let a differentiable function $f$ satisfy $f(x) + \int_{3}^{x} \frac{f(t)}{t} dt = \sqrt{x+1}, x \ge 3$. Then $12f(8)$ is equal to:

Options:

(A) $34$

(B) $19$

(C) $17$

(D) $1$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given the functional equation:
$$f(x) + \int_{3}^{x} \frac{f(t)}{t} dt = \sqrt{x+1}$$

Our mission is to determine the value of $12f(8)$. The presence of the function $f(t)$ inside an integral with a variable upper limit suggests that we should apply the **Newton-Leibniz rule** to transform this into a differential equation.

### The Liberation

To free the function, we differentiate both sides with respect to $x$. Applying the Leibniz rule to the integral term, we obtain:

$$f'(x) + \frac{f(x)}{x} = \frac{1}{2\sqrt{x+1}}$$

This is a classic **first-order Linear Differential Equation (LDE)** of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = \frac{1}{x}$ and $Q(x) = \frac{1}{2\sqrt{x+1}}$.

### The Architecture

To solve this LDE, we calculate the Integrating Factor (I.F.):

$$I.F. = e^{\int P(x) dx} = e^{\int \frac{1}{x} dx} = e^{\ln x} = x$$

Multiplying the entire differential equation by the I.F. ($x$), the left side becomes the derivative of the product $x \cdot f(x)$:

$$\frac{d}{dx} [x \cdot f(x)] = \frac{x}{2\sqrt{x+1}}$$

### The Integration

Integrating both sides with respect to $x$, we have:

$$x \cdot f(x) = \int \frac{x}{2\sqrt{x+1}} dx$$

Using the substitution trick $x = (x+1) - 1$, the integral becomes:

$$x \cdot f(x) = \frac{1}{2} \int \left( \sqrt{x+1} - \frac{1}{\sqrt{x+1}} ight) dx$$

Performing the integration, we get:

$$x \cdot f(x) = \frac{1}{2} \left[ \frac{2}{3}(x+1)^{3/2} - 2\sqrt{x+1} ight] + C = \frac{(x+1)^{3/2}}{3} - \sqrt{x+1} + C$$

### The Final Victory

To find $C$, we use the original equation at $x=3$. Since the integral from $3$ to $3$ is zero, we have $f(3) = \sqrt{3+1} = 2$. Substituting these values into our general solution:

$$3(2) = \frac{4^{3/2}}{3} - \sqrt{4} + C \Rightarrow 6 = \frac{8}{3} - 2 + C \Rightarrow C = \frac{16}{3}$$

Now, we substitute $x=8$ to find $f(8)$:

$$8 f(8) = \frac{9^{3/2}}{3} - \sqrt{9} + \frac{16}{3} = 9 - 3 + \frac{16}{3} = 6 + \frac{16}{3} = \frac{34}{3}$$

Thus, $f(8) = \frac{34}{24} = \frac{17}{12}$. The final required value is:

$$12f(8) = 12 	imes \frac{17}{12} = \mathbf{17}$$

Animated Solution for Mathematics - Differential Equations: Let a differentiable function $f$ satisfy $f (x) + \int_{3}^{x} \frac{f ( t )}{t} d t = x + 1, x \geq 3$ . Then $12 f (8)$ is equal to:

Select Answer:

Visualized Solution

The Functional Equation

Applying Leibniz Rule

Identifying the L.D.E.

Calculating Integrating Factor

The General Solution Setup

Solving the Integral

Finding the Initial Condition

Solving for Constant C

Calculating f(8)

The Final Answer

The Sigma Insight: Linear Differential Equations

Analyzing the Setup

The Liberation

The Architecture

The Integration

The Final Victory

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Animated Solution for Mathematics - Differential Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a differentiable function f satisfy f(x)+∫3x​tf(t)​dt=x+1​,x≥3. Then 12f(8) is equal to: