`int_(0)^(pi//2) cos 3x dx`

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{2}} \cos(3x) \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ \int_{0}^{\frac{\pi}{2}} \cos(3x) \, dx \]

### Step 2: Find the antiderivative
The antiderivative of \( \cos(3x) \) can be found using the formula for the integral of cosine:
\[ \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \]
In our case, \( k = 3 \), so:
\[ \int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C \]

### Step 3: Evaluate the definite integral
Now we evaluate the definite integral from \( 0 \) to \( \frac{\pi}{2} \):
\[ \int_{0}^{\frac{\pi}{2}} \cos(3x) \, dx = \left[ \frac{1}{3} \sin(3x) \right]_{0}^{\frac{\pi}{2}} \]

### Step 4: Substitute the limits
We substitute the upper limit \( \frac{\pi}{2} \) and the lower limit \( 0 \):
\[ = \frac{1}{3} \sin\left(3 \cdot \frac{\pi}{2}\right) - \frac{1}{3} \sin(3 \cdot 0) \]
This simplifies to:
\[ = \frac{1}{3} \sin\left(\frac{3\pi}{2}\right) - \frac{1}{3} \sin(0) \]

### Step 5: Calculate the sine values
Now we calculate the sine values:
\[ \sin\left(\frac{3\pi}{2}\right) = -1 \quad \text{and} \quad \sin(0) = 0 \]
Substituting these values back, we get:
\[ = \frac{1}{3}(-1) - \frac{1}{3}(0) = -\frac{1}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} \cos(3x) \, dx = -\frac{1}{3} \]

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