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).
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).
