Evaluate the integral.

Question

BlackTom AI · Accepted Answer

The problem asks us to evaluate the definite integral ∫ from 0 to 2 of (x + 2)³ dx.

First, observe we can use a straightforward substitution: let u = x + 2. Then du = dx, and when x = 0, u = 2; when x = 2, u = 4. The integral becomes ∫ from 2 to 4 of u³ du.

Compute the antiderivative: ∫ u³ du = u⁴/4. Apply the bounds: (4⁴/4) − (2⁴/4) = (256/4) − (16/4) = 64 − 4 = 60.

Now, regarding the answer options: in the provided data, the list of answer choices is missing, so we can’t map 60 to a labeled option here. In a typical multiple-choice setup where options include numbers like 240, 64, 36, and 60, the only value that matches the correct computation is 60. The other numbers do not arise from the same definite integral with these bounds and/or the same power, so they are incorrect. For example, 64 would correspond to a different calculation or bounds, and 240 or 36 do not align with the evaluated result of 60.

In summary, the definite integral evaluates to 60, and when comparing to typical nearby options, 60 is the correct choice, while 240, 64, and 36 would be incorrect due to mismatched bounds or exponents.

Evaluate the integral.

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