What is `int_0^(pi//2) (cos x)dx` equal to

Options

A. A. `1/2`
B. B. 1
C. C. 2I
D. D. 4I

Answer

To solve the integral \( \int_0^{\frac{\pi}{2}} \cos x \, dx \), we can follow these steps:

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

### Step 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.

### Step 3: Apply the limits of integration
Now we will evaluate the definite integral by applying 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 upper limit
Substituting the upper limit \( \frac{\pi}{2} \):
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

### Step 5: Substitute the lower limit
Now substituting the lower limit \( 0 \):
\[ \sin(0) = 0 \]

### Step 6: Calculate the result
Now we can calculate the value of the integral:
\[ \int_0^{\frac{\pi}{2}} \cos x \, dx = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

### Final Answer
Thus, the value of the integral \( \int_0^{\frac{\pi}{2}} \cos x \, dx \) is \( 1 \).

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