The value of `int_(0)^(x) d[t]`, is

Options

A. x
B. `x^(2)//2`
C. `-x`
D. `[x]`

Explanation

To solve the integral \( I = \int_{0}^{x} dt \), we can follow these steps:

### Step 1: Understand the Integral
The integral \( \int_{0}^{x} dt \) represents the area under the curve of the function \( f(t) = 1 \) from \( t = 0 \) to \( t = x \).

### Step 2: Evaluate the Integral
Since the integrand is a constant function (1), we can directly integrate it:
\[ I = \int_{0}^{x} 1 \, dt \]

### Step 3: Apply the Fundamental Theorem of Calculus
The integral of a constant \( c \) over the interval \([a, b]\) is given by:
\[ \int_{a}^{b} c \, dt = c(b - a) \]
In our case, \( c = 1 \), \( a = 0 \), and \( b = x \):
\[ I = 1 \cdot (x - 0) = x \]

### Step 4: Write the Final Answer
Thus, the value of the integral is:
\[ I = x \]