If `int_0^(pi//3) . ("cos x")/(3 + 4 "sin " x) dx = k "log" ((3 + 2sqrt(3))/(3))` then k is

Options

A. `1/2`
B. `1/3`
C. `1/4`
D. `1/8`

Answer

To solve the integral

\[ \int_0^{\frac{\pi}{3}} \frac{\cos x}{3 + 4 \sin x} \, dx = k \log\left(\frac{3 + 2\sqrt{3}}{3}\right), \]

we will follow these steps:

### Step 1: Substitution
Let \( t = 3 + 4 \sin x \). Then, we differentiate \( t \):
\[ dt = 4 \cos x \, dx \quad \Rightarrow \quad dx = \frac{dt}{4 \cos x}. \]
Also, we need to change the limits of integration. When \( x = 0 \):
\[ t = 3 + 4 \sin(0) = 3. \]
When \( x = \frac{\pi}{3} \):
\[ t = 3 + 4 \sin\left(\frac{\pi}{3}\right) = 3 + 4 \cdot \frac{\sqrt{3}}{2} = 3 + 2\sqrt{3}. \]

### Step 2: Rewrite the Integral
Now, substituting \( t \) into the integral:
\[ \int_3^{3 + 2\sqrt{3}} \frac{\cos x}{t} \cdot \frac{dt}{4 \cos x} = \frac{1}{4} \int_3^{3 + 2\sqrt{3}} \frac{1}{t} \, dt. \]

### Step 3: Evaluate the Integral
The integral of \( \frac{1}{t} \) is:
\[ \int \frac{1}{t} \, dt = \log |t|. \]
Thus, we have:
\[ \frac{1}{4} \left[ \log |t| \right]_3^{3 + 2\sqrt{3}} = \frac{1}{4} \left( \log(3 + 2\sqrt{3}) - \log(3) \right). \]
Using the property of logarithms \( \log a - \log b = \log \left( \frac{a}{b} \right) \):
\[ = \frac{1}{4} \log \left( \frac{3 + 2\sqrt{3}}{3} \right). \]

### Step 4: Compare with Given Expression
We have:
\[ \frac{1}{4} \log \left( \frac{3 + 2\sqrt{3}}{3} \right) = k \log \left( \frac{3 + 2\sqrt{3}}{3} \right). \]
Dividing both sides by \( \log \left( \frac{3 + 2\sqrt{3}}{3} \right) \) (assuming it is non-zero):
\[ \frac{1}{4} = k. \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{4}}. \]