`int_(0)^(oo)[2e^(-x)]dx`, where `[.]` deontes greatest integer function, is equal to

Options

A. 0
B. `ln2`
C. `e^(2)`
D. `2e^(-1)`

Answer

To solve the integral \( \int_{0}^{\infty} 2e^{-x} \, dx \) and find the value of the greatest integer function, we can follow these steps:

### Step 1: Evaluate the integral

The integral we need to evaluate is:

\[ \int_{0}^{\infty} 2e^{-x} \, dx \]

We can factor out the constant 2:

\[ = 2 \int_{0}^{\infty} e^{-x} \, dx \]

### Step 2: Solve the integral \( \int_{0}^{\infty} e^{-x} \, dx \)

The integral \( \int_{0}^{\infty} e^{-x} \, dx \) can be calculated as follows:

\[ \int e^{-x} \, dx = -e^{-x} + C \]

Evaluating the definite integral from 0 to \( \infty \):

\[ \left[-e^{-x}\right]_{0}^{\infty} = \lim_{b \to \infty} (-e^{-b}) - (-e^{0}) = 0 - (-1) = 1 \]

### Step 3: Substitute back into the integral

Now substituting back into our original integral:

\[ 2 \int_{0}^{\infty} e^{-x} \, dx = 2 \cdot 1 = 2 \]

### Step 4: Apply the greatest integer function

The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). Since we have calculated the integral to be 2, we apply the greatest integer function:

\[ [2] = 2 \]

### Final Answer

Thus, the value of the integral \( \int_{0}^{\infty} 2e^{-x} \, dx \) when applied to the greatest integer function is:

\[ \boxed{2} \]

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