Evaluate the Integral

Problem

(∫_0^ln(2))(e(2*x)*d(x))

Solution

1. Identify the integral form as a definite integral of an exponential function e(a*x)

2. Apply the power rule for integration of exponentials, where (∫_^)(e(a*x)*d(x))=(e(a*x))/a

3. Find the antiderivative of e(2*x) by dividing by the coefficient of x in the exponent.

(∫_^)(e(2*x)*d(x))=(e(2*x))/2

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

[(e(2*x))/2]ln(2)0

5. Substitute the upper limit into the expression.

(e(2*ln(2)))/2

6. Simplify the exponent using the property n*ln(a)=ln(an)

(eln(2))/2=(eln(4))/2

7. Evaluate the exponential of the natural log using eln(x)=x

4/2=2

8. Substitute the lower limit into the expression.

(e(2*(0)))/2=(e0)/2=1/2

9. Subtract the lower limit value from the upper limit value.

2−1/2=3/2

Final Answer

(∫_0^ln(2))(e(2*x)*d(x))=3/2

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