Value of `int_(0)^(pi//2) cos 3t` is

Options

A. `(2)/(3)`
B. `-(1)/(3)`
C. `- (2)/(3)`
D. `(1)/(3)`

Explanation

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

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

### Step 2: Use substitution
Let us use the substitution \( u = 3t \). Then, we differentiate to find \( du \):
\[ du = 3 \, dt \quad \Rightarrow \quad dt = \frac{du}{3} \]

### Step 3: Change the limits of integration
When \( t = 0 \):
\[ u = 3 \cdot 0 = 0 \]
When \( t = \frac{\pi}{2} \):
\[ u = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \]
Thus, the new limits for \( u \) are from \( 0 \) to \( \frac{3\pi}{2} \).

### Step 4: Substitute into the integral
Now we substitute \( u \) and \( dt \) into the integral:
\[ I = \int_{0}^{\frac{3\pi}{2}} \cos(u) \cdot \frac{du}{3} \]
This simplifies to:
\[ I = \frac{1}{3} \int_{0}^{\frac{3\pi}{2}} \cos(u) \, du \]

### Step 5: Integrate \( \cos(u) \)
The integral of \( \cos(u) \) is \( \sin(u) \):
\[ I = \frac{1}{3} \left[ \sin(u) \right]_{0}^{\frac{3\pi}{2}} \]

### Step 6: Evaluate the limits
Now we evaluate the sine function at the limits:
\[ I = \frac{1}{3} \left[ \sin\left(\frac{3\pi}{2}\right) - \sin(0) \right] \]
We know that:
\[ \sin\left(\frac{3\pi}{2}\right) = -1 \quad \text{and} \quad \sin(0) = 0 \]
Thus:
\[ I = \frac{1}{3} \left[ -1 - 0 \right] = \frac{-1}{3} \]

### Final Answer
The value of the integral \( \int_{0}^{\frac{\pi}{2}} \cos(3t) \, dt \) is:
\[ \boxed{-\frac{1}{3}} \]