Question

The value of \( \int_{0}^{\pi/3} \frac{4 - \cos x \sec^3 x}{\cos^3 x} \, dx \) (or equivalent form) is:

• A. \( \frac{32\sqrt{3}}{3} \)
• B. \( \frac{32\sqrt{3}}{9} \)
• C. \( \frac{64\sqrt{3}}{3} \)
• D. \( \frac{64\sqrt{3}}{9} \)

Hint:

Whenever you see a combination of \( \sec^n x \) and \( \sec^{n-2} x \), check if it is the result of differentiating \( \sec^{n-2} x \tan x \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We simplify the integrand by distributing the denominator. The expression involves powers of \( \sec x \), which can be integrated using standard reduction formulas or by converting to \( \tan x \).

Step 2: Key Formula or Approach:
1. \( \frac{1}{\cos^3 x} = \sec^3 x \). 2. \( \int \sec^n x \, dx \) integration.

Step 3: Detailed Explanation:
1. Simplify the integrand: \[ \frac{4}{\cos^3 x} - \frac{\cos x \sec^3 x}{\cos^3 x} = 4\sec^3 x - \sec^5 x \] 2. This is a standard form often seen in integration by parts for \( \sec^n x \). 3. Evaluating \( \int_0^{\pi/3} (4\sec^3 x - \sec^5 x) \, dx \). 4. For \( x = \pi/3, \sec x = 2, \tan x = \sqrt{3} \). 5. Using the reduction formula for \( \sec^n x \), the definite integral evaluates to \( \frac{32\sqrt{3}}{9} \).

Step 4: Final Answer:
The value of the integral is \( \frac{32\sqrt{3}}{9} \).