`int_(0)^(pi//2) cosx\ e^(sinx)\ dx` is equal to

Options

A. `e+1`
B. `e-1`
C. `e`
D. `-e`

Explanation

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \cos x \, e^{\sin x} \, dx \), we will use the substitution method. Here are the steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \cos x \, e^{\sin x} \, dx \]

### Step 2: Use substitution
Let us use the substitution:
\[ t = \sin x \]
Then, the derivative of \( t \) with respect to \( x \) is:
\[ dt = \cos x \, dx \]
This means that:
\[ \cos x \, dx = dt \]

### Step 3: Change the limits of integration
Now we need to change the limits of integration according to our substitution:
- When \( x = 0 \), \( t = \sin(0) = 0 \)
- When \( x = \frac{\pi}{2} \), \( t = \sin\left(\frac{\pi}{2}\right) = 1 \)

So, the new limits of integration are from \( 0 \) to \( 1 \).

### Step 4: Substitute in the integral
Now we can rewrite the integral in terms of \( t \):
\[ I = \int_{0}^{1} e^{t} \, dt \]

### Step 5: Evaluate the integral
The integral of \( e^{t} \) is:
\[ \int e^{t} \, dt = e^{t} \]
Thus, we evaluate it from \( 0 \) to \( 1 \):
\[ I = \left[ e^{t} \right]_{0}^{1} = e^{1} - e^{0} = e - 1 \]

### Final Result
Therefore, the value of the integral is:
\[ I = e - 1 \]