`int_(0)^(pi//2) e^(sin)x cos xdx` is equal to

Options

A. `e+1`
B. `e-1`
C. `e+2`
D. `e`

Explanation

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

### Step 1: Substitution
Let \( t = \sin x \). Then, the derivative of \( t \) with respect to \( x \) is:
\[ dt = \cos x \, dx \]
This means that \( dx = \frac{dt}{\cos x} \).

### Step 2: Change the Limits
Now we need to change the limits of integration. When \( x = 0 \):
\[ t = \sin(0) = 0 \]
When \( x = \frac{\pi}{2} \):
\[ t = \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus, the limits change from \( x = 0 \) to \( x = \frac{\pi}{2} \) into \( t = 0 \) to \( t = 1 \).

### Step 3: Rewrite the Integral
Now we can rewrite the integral in terms of \( t \):
\[ \int_{0}^{\frac{\pi}{2}} e^{\sin x} \cos x \, dx = \int_{0}^{1} e^{t} \, dt \]

### Step 4: Evaluate the Integral
The integral \(\int e^{t} \, dt\) is:
\[ e^{t} + C \]
Now we evaluate it from \( 0 \) to \( 1 \):
\[ \int_{0}^{1} e^{t} \, dt = \left[ e^{t} \right]_{0}^{1} = e^{1} - e^{0} = e - 1 \]

### Final Answer
Thus, the value of the integral \(\int_{0}^{\frac{\pi}{2}} e^{\sin x} \cos x \, dx\) is:
\[ e - 1 \]

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