Question

Evaluate the integral \[ \int_0^{\frac{\pi}{2}} \log \left( \frac{5 + 4 \sin x}{5 + 4 \cos x} \right) \, dx \]

• A. 0
• B. 2
• C. -2
• D. \( \frac{3}{4} \)

Hint:

When evaluating integrals involving symmetric functions, look for patterns or symmetries that can simplify the calculations, such as sine and cosine functions over symmetric intervals.

Answer 1

The Correct Option is A

Step 1: Use the property of logarithms.
We start by simplifying the given integral. Using the property of logarithms \( \log \left( \frac{a}{b} \right) = \log a - \log b \), we can rewrite the integrand:
\[ \int_0^{\frac{\pi}{2}} \left( \log(5 + 4 \sin x) - \log(5 + 4 \cos x) \right) \, dx \]

Step 2: Split the integral.
Now, we split the integral into two parts:
\[ \int_0^{\frac{\pi}{2}} \log(5 + 4 \sin x) \, dx - \int_0^{\frac{\pi}{2}} \log(5 + 4 \cos x) \, dx \]

Step 3: Symmetry of the integral.
Notice that the integrals are symmetric with respect to sine and cosine. In fact, by the symmetry of the interval \( [0, \frac{\pi}{2}] \), we know that:
\[ \int_0^{\frac{\pi}{2}} \log(5 + 4 \sin x) \, dx = \int_0^{\frac{\pi}{2}} \log(5 + 4 \cos x) \, dx \]

Step 4: Conclusion.
Since both integrals are equal, we subtract them:
\[ \int_0^{\frac{\pi}{2}} \log(5 + 4 \sin x) \, dx - \int_0^{\frac{\pi}{2}} \log(5 + 4 \cos x) \, dx = 0 \]

Step 5: Final Answer.
Therefore, the value of the integral is \( 0 \).