If `int_(0)^((pi)/(3))cos^(4)xdx=a pi+b sqrt(3)` ,where "a" and "b" are rational numbers, then `9a+8b` is equal to:

Options

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

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx \) and express it in the form \( a\pi + b\sqrt{3} \), we will follow these steps:

### Step 1: Rewrite \( \cos^4 x \)
We can use the identity for \( \cos^2 x \):
\[ \cos^2 x = \frac{1 + \cos 2x}{2} \]
Thus,
\[ \cos^4 x = \left(\cos^2 x\right)^2 = \left(\frac{1 + \cos 2x}{2}\right)^2 = \frac{(1 + \cos 2x)^2}{4} \]

### Step 2: Expand \( \cos^4 x \)
Expanding \( \cos^4 x \):
\[ \cos^4 x = \frac{1 + 2\cos 2x + \cos^2 2x}{4} \]
Now, we can express \( \cos^2 2x \) using the same identity:
\[ \cos^2 2x = \frac{1 + \cos 4x}{2} \]
Substituting this back, we have:
\[ \cos^4 x = \frac{1 + 2\cos 2x + \frac{1 + \cos 4x}{2}}{4} = \frac{1 + 2\cos 2x + \frac{1}{2} + \frac{1}{2}\cos 4x}{4} \]
Combining terms gives:
\[ \cos^4 x = \frac{3/2 + 2\cos 2x + \frac{1}{2}\cos 4x}{4} = \frac{3 + 4\cos 2x + \cos 4x}{8} \]

### Step 3: Set up the integral
Now we can rewrite the integral:
\[ \int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = \int_{0}^{\frac{\pi}{3}} \frac{3 + 4\cos 2x + \cos 4x}{8} \, dx = \frac{1}{8} \int_{0}^{\frac{\pi}{3}} (3 + 4\cos 2x + \cos 4x) \, dx \]

### Step 4: Evaluate the integral
Now we will evaluate each part of the integral:
1. \( \int_{0}^{\frac{\pi}{3}} 3 \, dx = 3 \cdot \frac{\pi}{3} = \pi \)
2. \( \int_{0}^{\frac{\pi}{3}} 4\cos 2x \, dx = 4 \cdot \frac{\sin 2x}{2} \bigg|_{0}^{\frac{\pi}{3}} = 2(\sin \frac{2\pi}{3} - \sin 0) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \)
3. \( \int_{0}^{\frac{\pi}{3}} \cos 4x \, dx = \frac{\sin 4x}{4} \bigg|_{0}^{\frac{\pi}{3}} = \frac{\sin \frac{4\pi}{3} - \sin 0}{4} = \frac{-\frac{\sqrt{3}}{2}}{4} = -\frac{\sqrt{3}}{8} \)

Putting it all together:
\[ \int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = \frac{1}{8} \left( \pi + \sqrt{3} - \frac{\sqrt{3}}{8} \right) = \frac{1}{8} \left( \pi + \frac{8\sqrt{3}}{8} - \frac{\sqrt{3}}{8} \right) = \frac{1}{8} \left( \pi + \frac{7\sqrt{3}}{8} \right) \]

### Step 5: Express in the required form
Thus, we have:
\[ \int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = \frac{1}{8} \pi + \frac{7}{64} \sqrt{3} \]
From this, we can identify \( a = \frac{1}{8} \) and \( b = \frac{7}{64} \).

### Step 6: Calculate \( 9a + 8b \)
Now we calculate:
\[ 9a + 8b = 9 \cdot \frac{1}{8} + 8 \cdot \frac{7}{64} = \frac{9}{8} + \frac{56}{64} = \frac{9}{8} + \frac{7}{8} = \frac{16}{8} = 2 \]

### Final Answer
Thus, the value of \( 9a + 8b \) is \( \boxed{2} \).