Key Concepts and Formulas

• Leibniz Integral Rule (Fundamental Theorem of Calculus Part I): If $F(x) = \int_{a}^{x} g(t) dt$, where $a$ is a constant, then $F'(x) = g(x)$.
• First-Order Linear Differential Equation: An equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$. The integrating factor is $IF = e^{\int P(x) dx}$, and the general solution is $y \cdot IF = \int Q(x) \cdot IF \ dx + C$.
• Substitution Method for Integration: A technique to simplify integrals by changing the variable of integration.

Step-by-Step Solution

1. Differentiate the Given Equation

We are given the equation: $f(x)+\int_{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, \quad x \geq 3$

We differentiate both sides of the equation with respect to $x$ to eliminate the integral.

• $\frac{d}{dx}[f(x)] = f'(x)$
• $\frac{d}{dx}\left[\int_{3}^{x} \frac{f(t)}{t} d t\right] = \frac{f(x)}{x}$ (using the Leibniz Integral Rule)
• $\frac{d}{dx}[\sqrt{x+1}] = \frac{d}{dx}[(x+1)^{1/2}] = \frac{1}{2}(x+1)^{-1/2} = \frac{1}{2\sqrt{x+1}}$

Combining these, we get the differential equation: $f'(x) + \frac{f(x)}{x} = \frac{1}{2\sqrt{x+1}}$

Why this step? Differentiating the integro-differential equation transforms it into a standard differential equation, which we can then solve using standard techniques.

2. Identify the Form of the Differential Equation

Let $y = f(x)$, so $f'(x) = \frac{dy}{dx}$. Substituting this into the equation: $\frac{dy}{dx} + \frac{1}{x}y = \frac{1}{2\sqrt{x+1}}$

This is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where:

• $P(x) = \frac{1}{x}$
• $Q(x) = \frac{1}{2\sqrt{x+1}}$

Why this step? Recognizing the standard form allows us to apply the method of integrating factors, a well-established technique for solving linear differential equations.

3. Calculate the Integrating Factor (IF)

The integrating factor is calculated using the formula $IF = e^{\int P(x) dx}$. Substitute $P(x) = \frac{1}{x}$: $IF = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = x$ Since $x \geq 3$, we have $|x| = x$.

Why this step? The integrating factor will simplify the left-hand side of the equation into the derivative of a product, making the equation directly integrable.

4. Find the General Solution of the Differential Equation

The general solution is given by $y \cdot IF = \int Q(x) \cdot IF \ dx + C$. Substitute $y = f(x)$, $IF = x$, and $Q(x) = \frac{1}{2\sqrt{x+1}}$: $x f(x) = \int \frac{x}{2\sqrt{x+1}} dx + C = \frac{1}{2} \int \frac{x}{\sqrt{x+1}} dx + C$

Why this step? This step sets up the integration that will lead to the general solution.

5. Solve the Integral $\int \frac{x}{\sqrt{x+1}} dx$

We use the substitution method. Let $u = x+1$, so $x = u-1$ and $du = dx$. $\int \frac{x}{\sqrt{x+1}} dx = \int \frac{u-1}{\sqrt{u}} du = \int \left(\frac{u}{\sqrt{u}} - \frac{1}{\sqrt{u}}\right) du = \int (u^{1/2} - u^{-1/2}) du$ Integrating each term: $= \frac{u^{3/2}}{3/2} - \frac{u^{1/2}}{1/2} + C' = \frac{2}{3}u^{3/2} - 2u^{1/2} + C'$ Substitute back $u = x+1$: $= \frac{2}{3}(x+1)^{3/2} - 2(x+1)^{1/2} + C'$ Now substitute this back into the general solution from Step 4: $x f(x) = \frac{1}{2} \left( \frac{2}{3}(x+1)^{3/2} - 2(x+1)^{1/2} \right) + C = \frac{1}{3}(x+1)^{3/2} - (x+1)^{1/2} + C$

Why this step? Accurate integration is crucial. The substitution simplifies the integral, allowing us to evaluate it using the power rule.

6. Determine the Constant of Integration (C)

From the original equation, $f(x)+\int_{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}$. Substitute $x = 3$: $f(3)+\int_{3}^{3} \frac{f(t)}{t} d t=\sqrt{3+1} = \sqrt{4} = 2$ Since $\int_{3}^{3} \frac{f(t)}{t} d t = 0$, we have $f(3) = 2$. Now substitute $x=3$ and $f(3)=2$ into the general solution: $3 \cdot 2 = \frac{1}{3}(3+1)^{3/2} - (3+1)^{1/2} + C$ $6 = \frac{1}{3}(4)^{3/2} - (4)^{1/2} + C = \frac{1}{3}(8) - 2 + C = \frac{8}{3} - 2 + C = \frac{2}{3} + C$ $C = 6 - \frac{2}{3} = \frac{18}{3} - \frac{2}{3} = \frac{16}{3}$

Why this step? The initial condition, found by using the lower limit of integration in the original equation, allows us to determine the particular solution.

7. Write Down the Particular Solution for $f(x)$

We have $C = \frac{16}{3}$. Substituting this into our general solution: $x f(x) = \frac{1}{3}(x+1)^{3/2} - (x+1)^{1/2} + \frac{16}{3}$ Dividing by $x$: $f(x) = \frac{1}{x} \left( \frac{1}{3}(x+1)^{3/2} - (x+1)^{1/2} + \frac{16}{3} \right)$

Why this step? This is the explicit form of the function $f(x)$ that satisfies all the conditions.

8. Evaluate $12f(8)$

We need to find $f(8)$. Substitute $x = 8$: $8 f(8) = \frac{1}{3}(8+1)^{3/2} - (8+1)^{1/2} + \frac{16}{3} = \frac{1}{3}(9)^{3/2} - (9)^{1/2} + \frac{16}{3}$ $8 f(8) = \frac{1}{3}(27) - 3 + \frac{16}{3} = 9 - 3 + \frac{16}{3} = 6 + \frac{16}{3} = \frac{18}{3} + \frac{16}{3} = \frac{34}{3}$ Now find $12 f(8)$: $12 f(8) = \frac{12}{8} \cdot \frac{34}{3} = \frac{3}{2} \cdot \frac{34}{3} = \frac{34}{2} = 17$

Why this step? This is the final step to calculate the desired value.

Common Mistakes & Tips

• Forgetting the constant of integration: Always remember to add the constant of integration after performing an indefinite integral.
• Incorrectly applying the Leibniz rule: Ensure the limits of integration are correctly handled when differentiating.
• Algebraic errors: Pay close attention to algebraic manipulations, especially when dealing with fractions and exponents.

Summary

We solved an integro-differential equation by differentiating it to obtain a first-order linear differential equation. We then found the integrating factor, solved the resulting integral, and used the initial condition derived from the original equation to determine the constant of integration. Finally, we substituted $x=8$ into the particular solution to find $f(8)$ and calculate $12f(8)$.

The final answer is $\boxed{17}$, which corresponds to option (C).