Evaluate the Integral

Problem

(∫_0^ln(2))(e^x*d(x))

Solution

1. Identify the integral as a definite integral of the exponential function e^x

2. Find the antiderivative of e^x which is simply e^x

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit ln(2) and the lower limit 0

[e^x]^ln(2)_0

4. Substitute the limits into the expression.

e^ln(2)−e^0

5. Simplify the terms using the properties of logarithms and exponents, noting that e^ln(2)=2 and e^0=1

2−1

6. Calculate the final difference.

1

Final Answer

(∫_0^ln(2))(e^x*d(x))=1

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