The exact value of $ \int_{-2}^{2}({4e^{2x}-2e^{-2x})dx} $ is:

Question

BlackTom AI · Accepted Answer

Integrate term-by-term: the antiderivative of e^{ax} is (1/a)e^{ax}, so use that for each exponential.

1) Relevant concepts
- Linearity of the integral: ∫(A+B) = ∫A + ∫B.
- ∫ e^{ax} dx = (1/a)e^{ax} + C.
- Evaluate a definite integral by computing F(b) − F(a) where F is an antiderivative.

2) Step-by-step reasoning and intermediate calculations
Start with the integrand: 4e^{2x} − 2e^{-2x}.
Compute antiderivative F(x):
- For 4e^{2x}: ∫4e^{2x} dx = 4 · (1/2)e^{2x} = 2e^{2x}.
- For −2e^{-2x}: ∫−2e^{-2x} dx = −2 · (1/−2)e^{-2x} = +1·e^{-2x} = e^{-2x}.
So F(x) = 2e^{2x} + e^{-2x}.

Evaluate from x = −2 to x = 2:
F(2) = 2e^{4} + e^{-4}.
F(−2) = 2e^{-4} + e^{4}.
Now subtract: F(2) − F(−2) = (2e^{4} + e^{-4}) − (2e^{-4} + e^{4})
= 2e^{4} + e^{-4} − 2e^{-4} − e^{4}
= (2e^{4} − e^{4}) + (e^{-4} − 2e^{-4})
= e^{4} − e^{-4}.

3) Why this is correct
The antiderivative was obtained correctly using the standard exponential rule and the definite integral was evaluated exactly by direct substitution. The algebraic simplification yields e^{4} − e^{-4}.

4) Why the other choices are incorrect
- 2e^{4} − e^{-4}: corresponds to mis-evaluating subtraction of F(−2) (forgot a term).
- 3e^{4} − e^{-4}: arises from an arithmetic error combining coefficients.
- e^{4} + e^{-4}: would be the result if signs were added instead of subtracted.
Each of these mistakes misapplies the final subtraction or combines terms incorrectly.

Conclusion: The exact value is e^{4} − e^{-4}, which matches the provided correct answer (d).

The exact value of \( \int_{-2}^{2}({4e^{2x}-2e^{-2x})dx} \) is:单项选择题

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