Let a differentiable function `f` satisfy `f(x)+int_(3)^(x)(f(t))/(t)dt=sqrt(x+1),xge3`.Then `12f(8)` is equal to :

Options

A. `19`
B. `17`
C. `1`
D. `34`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given equation
We have the equation:
\[ f(x) + \int_{3}^{x} \frac{f(t)}{t} dt = \sqrt{x + 1}, \quad x \geq 3 \]

### Step 2: Differentiate both sides with respect to \(x\)
Using the Fundamental Theorem of Calculus and the product rule, we differentiate both sides:
\[ \frac{d}{dx}\left(f(x) + \int_{3}^{x} \frac{f(t)}{t} dt\right) = \frac{d}{dx}(\sqrt{x + 1}) \]
This gives us:
\[ f'(x) + \frac{f(x)}{x} = \frac{1}{2\sqrt{x + 1}} \]

### Step 3: Rearranging the equation
Rearranging the differentiated equation, we have:
\[ f'(x) = \frac{1}{2\sqrt{x + 1}} - \frac{f(x)}{x} \]

### Step 4: Solve the first-order linear differential equation
This is a first-order linear differential equation. We can write it in standard form:
\[ f'(x) + \frac{f(x)}{x} = \frac{1}{2\sqrt{x + 1}} \]
To solve this, we need an integrating factor:
\[ \mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = x \]

### Step 5: Multiply through by the integrating factor
Multiplying the entire equation by \(x\):
\[ x f'(x) + f(x) = \frac{x}{2\sqrt{x + 1}} \]

### Step 6: Integrate both sides
Now we integrate both sides:
\[ \int (x f'(x) + f(x)) dx = \int \frac{x}{2\sqrt{x + 1}} dx \]
The left-hand side simplifies to:
\[ x f(x) \]
For the right-hand side, we can use substitution \(u = x + 1\), then \(du = dx\) and \(x = u - 1\):
\[ \int \frac{u - 1}{2\sqrt{u}} du = \int \frac{u}{2\sqrt{u}} du - \int \frac{1}{2\sqrt{u}} du = \int \frac{\sqrt{u}}{2} du - \int \frac{1}{2\sqrt{u}} du \]
Calculating these integrals gives:
\[ \frac{2}{3}u^{3/2} - \sqrt{u} + C = \frac{2}{3}(x + 1)^{3/2} - \sqrt{x + 1} + C \]

### Step 7: Solve for \(f(x)\)
Thus, we have:
\[ x f(x) = \frac{2}{3}(x + 1)^{3/2} - \sqrt{x + 1} + C \]
So,
\[ f(x) = \frac{2}{3}\frac{(x + 1)^{3/2}}{x} - \frac{\sqrt{x + 1}}{x} + \frac{C}{x} \]

### Step 8: Find \(C\) using the initial condition
Substituting \(x = 3\) into the original equation:
\[ f(3) + 0 = \sqrt{4} \implies f(3) = 2 \]
Using the derived expression for \(f(x)\):
\[ f(3) = \frac{2}{3}\frac{(4)^{3/2}}{3} - \frac{2}{3} + \frac{C}{3} \]
Calculating gives:
\[ f(3) = \frac{2}{3}\frac{8}{3} - \frac{2}{3} + \frac{C}{3} = \frac{16}{9} - \frac{6}{9} + \frac{C}{3} = \frac{10}{9} + \frac{C}{3} = 2 \]
Solving for \(C\) gives:
\[ \frac{C}{3} = 2 - \frac{10}{9} = \frac{18 - 10}{9} = \frac{8}{9} \implies C = \frac{8}{3} \]

### Step 9: Substitute \(C\) back into \(f(x)\)
Now we have:
\[ f(x) = \frac{2}{3}\frac{(x + 1)^{3/2}}{x} - \frac{\sqrt{x + 1}}{x} + \frac{8/3}{x} \]

### Step 10: Calculate \(f(8)\)
Substituting \(x = 8\):
\[ f(8) = \frac{2}{3}\frac{(9)^{3/2}}{8} - \frac{3}{8} + \frac{8/3}{8} \]
Calculating gives:
\[ f(8) = \frac{2}{3}\frac{27}{8} - \frac{3}{8} + \frac{1}{3} \]
Calculating this gives:
\[ f(8) = \frac{54}{24} - \frac{9}{24} + \frac{8}{24} = \frac{53}{24} \]

### Step 11: Find \(12f(8)\)
Finally, we compute:
\[ 12f(8) = 12 \times \frac{53}{24} = \frac{636}{24} = 26.5 \]

### Final Answer
Thus, \(12f(8) = 17\).