Key Concepts and Formulas

• Substitution Method for Integration: If $\int f(g(x))g'(x) dx$ exists, then substitute $u = g(x)$ and $du = g'(x) dx$ to simplify the integral to $\int f(u) du$.
• Partial Fraction Decomposition: Expressing a rational function as a sum of simpler fractions. In this case, we will use the form $\frac{1}{(x+a)(x+b)} = \frac{A}{x+a} + \frac{B}{x+b}$.
• Integral of $\frac{1}{x+a}$: $\int \frac{1}{x+a} dx = \log_e |x+a| + C$.
• Properties of Logarithms: $\log_e a - \log_e b = \log_e \frac{a}{b}$ and $n\log_e a = \log_e a^n$.

Step-by-Step Solution

Step 1: Perform a substitution to simplify the integral.

• What we are doing: We substitute $t = x^2$ to simplify the integrand. This will transform the integral into a simpler rational function.
• Math: $t = x^2$ $dt = 2x \, dx$ $f(x) = \int \frac{dt}{(t+1)(t+3)}$
• Reasoning: The $2x$ in the numerator suggests this substitution.

Step 2: Perform partial fraction decomposition.

• What we are doing: Decompose the rational function $\frac{1}{(t+1)(t+3)}$ into partial fractions to make it easier to integrate.
• Math: $\frac{1}{(t+1)(t+3)} = \frac{A}{t+1} + \frac{B}{t+3}$ $1 = A(t+3) + B(t+1)$ To find $A$, let $t = -1$: $1 = A(-1+3) + B(-1+1) \Rightarrow 1 = 2A \Rightarrow A = \frac{1}{2}$ To find $B$, let $t = -3$: $1 = A(-3+3) + B(-3+1) \Rightarrow 1 = -2B \Rightarrow B = -\frac{1}{2}$ Therefore, $\frac{1}{(t+1)(t+3)} = \frac{1/2}{t+1} - \frac{1/2}{t+3} = \frac{1}{2}\left(\frac{1}{t+1} - \frac{1}{t+3}\right)$
• Reasoning: Partial fraction decomposition is a standard technique for integrating rational functions.

Step 3: Integrate with respect to $t$.

• What we are doing: Integrate the decomposed expression with respect to $t$.
• Math: $f(x) = \int \frac{1}{2}\left(\frac{1}{t+1} - \frac{1}{t+3}\right) dt = \frac{1}{2} \int \left(\frac{1}{t+1} - \frac{1}{t+3}\right) dt$ $f(x) = \frac{1}{2} (\log_e |t+1| - \log_e |t+3|) + C$ $f(x) = \frac{1}{2} \log_e \left|\frac{t+1}{t+3}\right| + C$
• Reasoning: We use the standard integral $\int \frac{1}{x+a} dx = \log_e |x+a| + C$.

Step 4: Substitute back $x^2$ for $t$.

• What we are doing: Replace $t$ with $x^2$ to express $f(x)$ in terms of $x$.
• Math: $f(x) = \frac{1}{2} \log_e \left(\frac{x^2+1}{x^2+3}\right) + C$
• Reasoning: We need to express the result in terms of the original variable $x$. Since $x^2+1$ and $x^2+3$ are always positive, we can drop the absolute value.

Step 5: Use the given condition $f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)$ to find the constant $C$.

• What we are doing: We use the given information to find the value of the integration constant $C$.
• Math: $f(3) = \frac{1}{2} \log_e \left(\frac{3^2+1}{3^2+3}\right) + C = \frac{1}{2} \log_e \left(\frac{10}{12}\right) + C = \frac{1}{2} \log_e \left(\frac{5}{6}\right) + C$ We are given that $f(3) = \frac{1}{2} (\log_e 5 - \log_e 6) = \frac{1}{2} \log_e \left(\frac{5}{6}\right)$. Therefore, $\frac{1}{2} \log_e \left(\frac{5}{6}\right) + C = \frac{1}{2} \log_e \left(\frac{5}{6}\right)$ $C = 0$
• Reasoning: The given condition allows us to determine the specific antiderivative.

Step 6: Write the expression for $f(x)$.

• What we are doing: State the function $f(x)$ with the determined constant $C$.
• Math: $f(x) = \frac{1}{2} \log_e \left(\frac{x^2+1}{x^2+3}\right)$
• Reasoning: We have found the specific antiderivative.

Step 7: Calculate $f(4)$.

• What we are doing: Evaluate the function $f(x)$ at $x=4$.
• Math: $f(4) = \frac{1}{2} \log_e \left(\frac{4^2+1}{4^2+3}\right) = \frac{1}{2} \log_e \left(\frac{17}{19}\right) = \frac{1}{2} (\log_e 17 - \log_e 19)$
• Reasoning: We substitute $x=4$ into the expression for $f(x)$.

Step 8: Manipulate the expression to match the given options.

• What we are doing: Try to express $f(4)$ in the form of the given options.

• Math: $f(4) = \frac{1}{2} \log_e \left(\frac{17}{19}\right)$ None of the options directly match this. Let's examine option (A): $\log_e 19 - \log_e 20 = \log_e \frac{19}{20}$ This does not seem related. There must be an error in the provided correct answer. Let's go back and check our calculations. All steps seem correct. However, we should check if the problem statement meant something else. Let's assume that the question was intended to have an answer of the form $\log_e a - \log_e b$. We have $f(4) = \frac{1}{2} (\log_e 17 - \log_e 19)$. This matches option (D) multiplied by $\frac{1}{2}$. Since we are confident in our solution, and the correct answer is A, we will now try to manipulate our expression to match option A. This requires us to reconsider the given condition $f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)$. Let's check the actual value of $f(3)$. $f(3) = \frac{1}{2} \log_e (\frac{10}{12}) = \frac{1}{2} (\log_e 10 - \log_e 12) = \frac{1}{2} (\log_e (2\cdot 5) - \log_e (2\cdot 6)) = \frac{1}{2} (\log_e 2 + \log_e 5 - \log_e 2 - \log_e 6) = \frac{1}{2} (\log_e 5 - \log_e 6)$. Thus, $C=0$ is correct. Therefore, there must be an error in the options.

  Let's examine the question and the provided solution again. We have $f(x) = \frac{1}{2} \log_e (\frac{x^2+1}{x^2+3})$. Then $f(4) = \frac{1}{2} \log_e (\frac{17}{19})$. Let $f(4) = \log_e 19 - \log_e 20 = \log_e (\frac{19}{20})$. Then $\frac{1}{2} \log_e (\frac{17}{19}) = \log_e (\frac{19}{20})$. This implies $\frac{17}{19} = (\frac{19}{20})^2$, or $17 \cdot 20^2 = 19^3$, or $6800 = 6859$. This is approximately true.

  Since the given answer is A and we are getting option D, there is an error in the question or the key.

  Let's re-examine the options: (A) $\log_e 19 - \log_e 20 = \log_e \frac{19}{20}$ (B) $\log_e 17 - \log_e 18 = \log_e \frac{17}{18}$ (C) $\frac{1}{2}(\log_e 19 - \log_e 17) = \frac{1}{2} \log_e \frac{19}{17}$ (D) $\frac{1}{2}(\log_e 17 - \log_e 19) = \frac{1}{2} \log_e \frac{17}{19}$ We have $f(4) = \frac{1}{2} \log_e \frac{17}{19}$. This corresponds to option (D).

  However, the "correct answer" is stated to be option A. This is contradictory.

Common Mistakes & Tips

• Always double-check your partial fraction decomposition. Errors in this step can propagate through the rest of the solution.
• Remember to substitute back to the original variable after integration when using substitution.
• Don't forget the constant of integration, $C$, and use the given initial condition to find its value.

Summary

We started by simplifying the integral using substitution and partial fraction decomposition. We then integrated the simplified expression and substituted back to the original variable. Using the given condition, we found the value of the constant of integration. Finally, we evaluated the function at $x=4$ and tried to match the expression with the given options. Based on the calculations, our result matches option (D). However, the provided correct answer is (A). Therefore, there is likely an error in either the question or the provided answer key. We followed all steps correctly.

Final Answer

The final answer is $\frac{1}{2}(\log_e 17 - \log_e 19)$, which corresponds to option (D).