Problem

True or false: By the Evaluation Theorem,
\[
\int_{-2}^{2} \frac{1}{x^{2}} d x=\left.\left(-\frac{1}{x}\right)\right|_{-2} ^{2}=-1
\]
True
False

Answer

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Answer

Final Answer: \(\boxed{\text{True}}\)

Steps

Step 1 :The Evaluation Theorem states that the definite integral of a function from a to b is equal to the antiderivative of the function evaluated at b minus the antiderivative of the function evaluated at a. In this case, the function is \(\frac{1}{x^{2}}\) and its antiderivative is \(-\frac{1}{x}\). So, we need to evaluate \(-\frac{1}{x}\) at x=2 and x=-2 and subtract the two results. If the result is -1, then the statement is true. Otherwise, it is false.

Step 2 :The result of the evaluation is -1, which matches the value given in the question. Therefore, the statement in the question is true.

Step 3 :Final Answer: \(\boxed{\text{True}}\)

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