Evaluate the following integrals
`int_(0)^(pi//2)cos x dx `

Answer

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

### Step-by-Step Solution:

1. **Identify the Integral**: We are asked to evaluate the definite integral of \( \cos x \) from \( 0 \) to \( \frac{\pi}{2} \).

\[ \int_{0}^{\frac{\pi}{2}} \cos x \, dx \]

2. **Find the Antiderivative**: The antiderivative of \( \cos x \) is \( \sin x \). Therefore, we can write:

\[ \int \cos x \, dx = \sin x + C \]

where \( C \) is the constant of integration.

3. **Apply the Limits**: Since we are dealing with a definite integral, we will evaluate \( \sin x \) at the upper limit \( \frac{\pi}{2} \) and the lower limit \( 0 \):

\[ \left[ \sin x \right]_{0}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) \]

4. **Calculate the Values**: Now, we substitute the values:

- \( \sin\left(\frac{\pi}{2}\right) = 1 \)
- \( \sin(0) = 0 \)

Therefore, we have:

\[ \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

5. **Final Result**: Thus, the value of the definite integral is:

\[ \int_{0}^{\frac{\pi}{2}} \cos x \, dx = 1 \]