The integral value of [tex]\(\int_0^1 e^x \, dx\)[/tex] is:

A. [tex]\(\frac{1}{e}\)[/tex]
B. [tex]\(\frac{1.5}{c}\)[/tex]
C. [tex]\(\frac{1 \cdot e}{c}\)[/tex]
D. [tex]\(\frac{e \cdot 1}{e}\)[/tex]



Answer :

To find the value of the integral [tex]\(\int_0^1 e^x \, dx\)[/tex], we will solve it step by step:

1. Identify the integral: We need to evaluate the definite integral [tex]\(\int_0^1 e^x \, dx\)[/tex].

2. Find the antiderivative: The antiderivative of [tex]\(e^x\)[/tex] is [tex]\(e^x\)[/tex], since the derivative of [tex]\(e^x\)[/tex] is [tex]\(e^x\)[/tex].

3. Evaluate the definite integral: Using the Fundamental Theorem of Calculus, we evaluate the antiderivative at the upper limit and lower limit, then subtract:
[tex]\[ \left[ e^x \right]_0^1 = e^1 - e^0 \][/tex]

4. Simplify the expression:
[tex]\[ e^1 - e^0 = e - 1 \][/tex]

Therefore, the value of [tex]\(\int_0^1 e^x \, dx\)[/tex] is [tex]\(e - 1\)[/tex].

Given the choices, none of the provided options directly matches [tex]\(e - 1\)[/tex]. Thus, the correct answer is not listed among the options provided (A, B, C, D).