`int_(pi//6)^(pi//3) sin(3x)dx=`

Options

A. `(1)/(3)`
B. `-(1)/(3)`
C. `1`
D. `3`

Explanation

To solve the integral \( \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin(3x) \, dx \), we will follow these steps:

### Step 1: Identify the integral
We need to compute the integral of \( \sin(3x) \) from \( \frac{\pi}{6} \) to \( \frac{\pi}{3} \).

### Step 2: Find the antiderivative
The integral of \( \sin(kx) \) is given by:
\[ \int \sin(kx) \, dx = -\frac{1}{k} \cos(kx) + C \]
For our case, \( k = 3 \):
\[ \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) + C \]

### Step 3: Apply the limits of integration
Now we will evaluate the definite integral:
\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin(3x) \, dx = \left[-\frac{1}{3} \cos(3x)\right]_{\frac{\pi}{6}}^{\frac{\pi}{3}} \]

### Step 4: Substitute the upper limit
First, we substitute the upper limit \( x = \frac{\pi}{3} \):
\[ -\frac{1}{3} \cos\left(3 \cdot \frac{\pi}{3}\right) = -\frac{1}{3} \cos(\pi) = -\frac{1}{3} \cdot (-1) = \frac{1}{3} \]

### Step 5: Substitute the lower limit
Now, we substitute the lower limit \( x = \frac{\pi}{6} \):
\[ -\frac{1}{3} \cos\left(3 \cdot \frac{\pi}{6}\right) = -\frac{1}{3} \cos\left(\frac{\pi}{2}\right) = -\frac{1}{3} \cdot 0 = 0 \]

### Step 6: Calculate the definite integral
Now we can combine the results from the upper and lower limits:
\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin(3x) \, dx = \frac{1}{3} - 0 = \frac{1}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sin(3x) \, dx = \frac{1}{3} \]

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