Question

If \( 3^a + 3^{-a} \), \( f(a) \) and \( 2^a + 2^{-a} \) are in A.P. If \( a \) is the minimum value of \( f(x) \), then the value of \( \int_{\ln 2}^{\ln 3} \frac{dx}{e^{2x} - e^{-2x}} \) is:

• A. \( \frac{1}{2} \ln \frac{4}{3} \)
• B. \( \frac{1}{4} \ln \frac{4}{3} \)
• C. \( \frac{1}{2} \ln \frac{8}{9} \)
• D. \( \frac{1}{4} \ln \frac{8}{9} \)

Hint:

When an integral contains \( e^{nx} \) and \( e^{-nx} \), multiplying the numerator and denominator by \( e^{nx} \) often transforms the problem into a standard substitution form \( u = e^{nx} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Since the three terms are in A.P., the middle term is the average of the extremes: \( f(a) = \frac{(3^a + 3^{-a}) + (2^a + 2^{-a})}{2} \). By AM-GM inequality, \( x^a + x^{-a} \ge 2 \). Therefore, the minimum value of \( f(a) \) occurs when \( a = 0 \), giving \( a = 0 \). The integral involves a hyperbolic-like denominator which can be solved using substitution.

Step 2: Key Formula or Approach:
1. \( f(a) = \frac{3^a + 3^{-a} + 2^a + 2^{-a}}{2} \). Min value occurs at \( a = 0 \). 2. For the integral, use substitution \( u = e^{2x} \), then \( du = 2e^{2x} dx \).

Step 3: Detailed Explanation:
1. The integral is \( I = \int_{\ln 2}^{\ln 3} \frac{dx}{e^{2x} - e^{-2x}} = \int_{\ln 2}^{\ln 3} \frac{e^{2x}}{e^{4x} - 1} dx \). 2. Let \( u = e^{2x} \). When \( x = \ln 2, u = e^{2 \ln 2} = 4 \). When \( x = \ln 3, u = e^{2 \ln 3} = 9 \). 3. \( du = 2e^{2x} dx \implies dx = \frac{du}{2u} \). 4. \( I = \int_{4}^{9} \frac{u}{u^2 - 1} \cdot \frac{du}{2u} = \frac{1}{2} \int_{4}^{9} \frac{du}{u^2 - 1} \). 5. Using the formula \( \int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln |\frac{u-a}{u+a}| \): \[ I = \frac{1}{2} \left[ \frac{1}{2} \ln \left| \frac{u-1}{u+1} \right| \right]_4^9 = \frac{1}{4} \left[ \ln \frac{8}{10} - \ln \frac{3}{5} \right] \] \[ I = \frac{1}{4} \ln \left( \frac{4/5}{3/5} \right) = \frac{1}{4} \ln \frac{4}{3} \]

Step 4: Final Answer:
The value of the integral is \( \frac{1}{4} \ln \frac{4}{3} \).