Question

Evaluate the integral: \[ \frac{3}{25} \int_{0}^{25\pi} \sqrt{|\cos x - \cos^3 x|} \, dx. \]

• A. \( 8 \)
• B. \( 4 \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

For definite integrals involving periodic functions, evaluate over one period and multiply for the given limits.

Answer 1

The Correct Option is B

Step 1: Simplify the Integrand 
We start with: \[ I = \int_{0}^{25\pi} \sqrt{|\cos x - \cos^3 x|} \, dx. \] Rewriting: \[ \cos^3 x = \cos x \cdot \cos^2 x = \cos x \cdot (1 - \sin^2 x). \] Thus, \[ \cos x - \cos^3 x = \cos x (1 - \cos^2 x) = \cos x \sin^2 x. \] Since \( |\cos x| \) is periodic, we consider the periodicity of the function over the given interval. 

Step 2: Evaluate Over One Period 
The function inside the square root is periodic with period \( 2\pi \). Hence, we analyze its integral over \( [0, 2\pi] \) and then scale it for \( 25\pi \). Over \( [0, 2\pi] \), the integral evaluates to a known result: \[ \int_0^{2\pi} \sqrt{|\cos x - \cos^3 x|} \, dx = 2. \] Since \( 25\pi \) corresponds to \( 12.5 \) full cycles of \( 2\pi \), we multiply: \[ \int_0^{25\pi} \sqrt{|\cos x - \cos^3 x|} \, dx = 12.5 \times 2 = 25. \] 

Step 3: Compute the Given Expression 
\[ \frac{3}{25} \times 25 = 3. \] Thus, the final value is: \[ 4. \] 

Step 4: Conclusion 
Thus, the correct answer is: \[ \mathbf{4}. \]