value of `int_0^((pi)/(2))cos3tdt ` 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 can follow these steps:

### Step 1: Use a substitution
Let \( x = 3t \). Then, differentiate both sides to find \( dt \):
\[ dx = 3 \, dt \quad \Rightarrow \quad dt = \frac{dx}{3} \]

### Step 2: Change the limits of integration
When \( t = 0 \):
\[ x = 3 \cdot 0 = 0 \]
When \( t = \frac{\pi}{2} \):
\[ x = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \]

### Step 3: Rewrite the integral
Now, substitute \( dt \) and change the limits:
\[ \int_0^{\frac{\pi}{2}} \cos(3t) \, dt = \int_0^{\frac{3\pi}{2}} \cos(x) \cdot \frac{dx}{3} \]
This simplifies to:
\[ \frac{1}{3} \int_0^{\frac{3\pi}{2}} \cos(x) \, dx \]

### Step 4: Evaluate the integral
The integral of \( \cos(x) \) is \( \sin(x) \):
\[ \frac{1}{3} \left[ \sin(x) \right]_0^{\frac{3\pi}{2}} = \frac{1}{3} \left( \sin\left(\frac{3\pi}{2}\right) - \sin(0) \right) \]

### Step 5: Calculate the sine values
We know:
\[ \sin\left(\frac{3\pi}{2}\right) = -1 \quad \text{and} \quad \sin(0) = 0 \]
Thus, we have:
\[ \frac{1}{3} \left( -1 - 0 \right) = \frac{-1}{3} \]

### Step 6: Final answer
The value of the integral \( \int_0^{\frac{\pi}{2}} \cos(3t) \, dt \) is:
\[ \frac{-1}{3} \]