`int_(0)^(pi//3) cos 3x dx=`

Options

A. `pi`
B. `0`
C. `(pi)/2`
D. `(pi)/4`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{3}} \cos(3x) \, dx \), we will follow these steps:

### Step 1: Identify the integral
We start with the integral:
\[ I = \int_{0}^{\frac{\pi}{3}} \cos(3x) \, dx \]

### Step 2: Use the integration formula
The integral of \( \cos(kx) \) is given by:
\[ \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \]
In our case, \( k = 3 \), so we have:
\[ \int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C \]

### Step 3: Apply the limits of integration
Now we will evaluate the definite integral from \( 0 \) to \( \frac{\pi}{3} \):
\[ I = \left[ \frac{1}{3} \sin(3x) \right]_{0}^{\frac{\pi}{3}} \]

### Step 4: Calculate the upper limit
First, we calculate the upper limit:
\[ \sin\left(3 \cdot \frac{\pi}{3}\right) = \sin(\pi) = 0 \]
Thus,
\[ \frac{1}{3} \sin(3 \cdot \frac{\pi}{3}) = \frac{1}{3} \cdot 0 = 0 \]

### Step 5: Calculate the lower limit
Next, we calculate the lower limit:
\[ \sin(3 \cdot 0) = \sin(0) = 0 \]
Thus,
\[ \frac{1}{3} \sin(3 \cdot 0) = \frac{1}{3} \cdot 0 = 0 \]

### Step 6: Combine the results
Now we substitute the limits back into the integral:
\[ I = 0 - 0 = 0 \]

### Final Answer
Thus, the value of the integral is:
\[ I = 0 \]

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