Evaluate the integral.

Question

BlackTom AI · Accepted Answer

To begin, I will restate the problem and the possible choices to ensure clarity.
Problem: Evaluate the definite integral ∫ from 0 to ln 5 of e^{2x} dx.
Possible answers (as seen): 24, 25/2, 12, 25.
Next, I examine the general method for integrating e^{ax}: the antiderivative of e^{ax} is (1/a) e^{ax}, so for e^{2x} it is (1/2) e^{2x}.
Now, I’ll apply the Fundamental Theorem of Calculus by evaluating this antiderivative at the bounds x = ln 5 and x = 0.
Compute the upper bound: (1/2) e^{2(ln 5)} = (1/2) e^{ln 25} = (1/2) * 25 = 25/2.
Compute the lower bound: (1/2) e^{2(0)} = (1/2) e^{0} = (1/2) * 1 = 1/2.
Subtract to find the definite integral: (25/2) − (1/2) = (24)/2 = 12.
Now, let’s assess each option:
- 24: This value is exactly twice the correct result. It would occur if one forgot the subtraction of the lower bound, i.e., if one computed (upper bound) only and ignored the minus (lower bound). This is a common mistake when applying the antiderivative to definite integrals.
- 25/2: This corresponds to the upper bound evaluation alone, forgetting to subtract the contribution from the lower limit x = 0. It reflects overlooking the subtraction step required by the definite integral.
- 12: This is the correct value after properly subtracting the lower limit from the upper limit, matching (25/2) − (1/2).
- 25: This outcome represents the upper bound value e^{2(ln 5)} = 25, i.e., it ignores the factor of 1/2 from integrating e^{2x}. It’s the result of misapplying the antiderivative without the 1/2 factor or misinterpreting the definite integral.
In summary, the correct reasoning yields 12, derived from the antiderivative (1/2) e^{2x} evaluated between 0 and ln 5, giving (25/2) − (1/2) = 12. The incorrect options arise from forgetting either the lower-bound subtraction or the 1/2 factor, or from confusing the upper bound with the definite integral result.

Evaluate the integral.

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