Evaluate the Integral

Problem

(∫_0^1)(e(−x)*d(x))

Solution

1. Identify the antiderivative of the function e(−x) using the rule (∫_^)(e(a*x)*d(x))=1/a*e(a*x)

2. Apply the rule with a=−1 to find the indefinite integral.

(∫_^)(e(−x)*d(x))=−e(−x)

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 1 and the lower limit 0

[−e(−x)]10

4. Substitute the limits into the expression.

−e(−1)−(−e0)

5. Simplify the expression using the properties e0=1 and e(−1)=1/e

−1/e+1

6. Rewrite the final result in a standard form.

1−1/e

Final Answer

(∫_0^1)(e(−x)*d(x))=1−1/e

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