Evaluate `int _(0)^(pi//2)cos x dx`

Answer

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

### Step 1: Identify the integral
We need to evaluate the integral of \( \cos x \) from \( 0 \) to \( \frac{\pi}{2} \).

### Step 2: Find the antiderivative of \( \cos x \)
The antiderivative (indefinite integral) of \( \cos x \) is:
\[ \int \cos x \, dx = \sin x + C \]
where \( C \) is the constant of integration.

### Step 3: Apply the limits of integration
Now we will evaluate the definite integral using the limits \( 0 \) and \( \frac{\pi}{2} \):
\[ \int_{0}^{\frac{\pi}{2}} \cos x \, dx = \left[ \sin x \right]_{0}^{\frac{\pi}{2}} \]

### Step 4: Substitute the limits into the antiderivative
We substitute the upper limit \( \frac{\pi}{2} \) and the lower limit \( 0 \):
\[ = \sin\left(\frac{\pi}{2}\right) - \sin(0) \]

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

### Final Answer
The value of the definite integral \( \int_{0}^{\frac{\pi}{2}} \cos x \, dx \) is:
\[ \boxed{1} \]