Choose the correct Answer of the Following Questions : `int_0^infty e^(-x) dx` is equal to

Options

A. -2
B. -7/16
C. 1/2
D. 1

Answer

To solve the definite integral \( \int_0^\infty e^{-x} \, dx \), we can follow these steps:

### Step 1: Identify the Integral
We need to evaluate the integral:
\[ \int_0^\infty e^{-x} \, dx \]

### Step 2: Find the Antiderivative
The antiderivative of \( e^{-x} \) is:
\[ -\frac{e^{-x}}{1} = -e^{-x} \]

### Step 3: Apply the Limits
Now we will apply the limits from 0 to infinity:
\[ \left[-e^{-x}\right]_0^\infty \]

### Step 4: Evaluate at the Upper Limit
First, we evaluate at the upper limit \( x = \infty \):
\[ -e^{-\infty} = -0 = 0 \]

### Step 5: Evaluate at the Lower Limit
Next, we evaluate at the lower limit \( x = 0 \):
\[ -e^{0} = -1 \]

### Step 6: Combine the Results
Now we combine the results from the limits:
\[ 0 - (-1) = 0 + 1 = 1 \]

### Conclusion
Thus, the value of the integral is:
\[ \int_0^\infty e^{-x} \, dx = 1 \]

### Final Answer
The correct answer is \( 1 \).

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