Integrate

∫
(
2
𝑥
−
1
)
4
𝑑
𝑥

Question

Integrate

∫
(
2
𝑥
−
1
)
4
𝑑
𝑥

BlackTom AI · Accepted Answer

We start by restating what’s given and what’s missing in the options.
- Question: Integrate ∫ (2x − 1)^4 dx.
- Answer options: not provided in the data (the field labeled answer_options is empty).
- The provided answer string appears to be intended as the result: (2x − 1)^5 / 10 + C, though it’s written as a concatenated and somewhat garbled form in the answer field.

To evaluate the integral, use a standard substitution technique. Let u = 2x − 1. Then du = 2 dx, so dx = du/2.
- The integral becomes ∫ (u^4) · (dx) = ∫ u^4 · (du/2) = (1/2) ∫ u^4 du.
- Integrating, (1/2) · (u^5 / 5) = u^5 / 10.
- Substituting back, the antiderivative is (2x − 1)^5 / 10 + C.

Now, considering the lack of answer options, there are a few possibilities:
- If one of the options were (2x − 1)^5 / 10 + C, that would be the correct choice, given the substitution work above.
- If an option attempted to present the result as (2x − 1)^5 divided by 10 without the constant term, that aligns with the standard indefinite integral, since constants are absorbed into C.
- Any option that misplaces the exponent (for example, (2x − 1)^4, or (2x − 1)^6) would be incorrect because it would not reflect the proper power progression from u^4 to u^5 when integrating with respect to u.

In summary, the correct antiderivative is (2x − 1)^5 / 10 + C. The absence of explicit answer choices in the provided data means we cannot point to a labeled correct option, but the derived form matches the standard integration procedure for ∫ (2x − 1)^4 dx.

If you obtain the actual list of answer options, I can map each one against this result and explain exactly which is correct and why the others are not.

Integrate ∫ ( 2 𝑥 − 1 ) 4 𝑑 𝑥

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