Evaluate the Integral

Problem

(∫_0^8)(e^(−x)*d(x))

Solution

1. Identify the antiderivative of the integrand 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 8 and the lower limit 0

[−e^(−x)]^8_0

4. Substitute the limits into the expression.

−e^(−8)−(−e^0)

5. Simplify the expression using the fact that e^0=1

−e^(−8)+1

6. Rearrange the terms for the final result.

1−e^(−8)

Final Answer

(∫_0^8)(e^(−x)*d(x))=1−e^(−8)

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