Find the integral

∫
𝑥
2
(
2
𝑥
3
+
1
)
4
𝑑
𝑥

Question

Find the integral

∫
𝑥
2
(
2
𝑥
3
+
1
)
4
𝑑
𝑥

BlackTom AI · Accepted Answer

The given integral appears as: ∫ x^2 (2x^3 + 1)^4 dx. However, the provided answer options are missing, so we cannot evaluate which choice matches among them. Here is a clear derivation of the antiderivative, which will help you verify any option if one were present.

Consider a substitution to simplify the integrand. Let u = 2x^3 + 1. Then du = 6x^2 dx, which gives x^2 dx = du/6.

Rewrite the integral in terms of u:
∫ x^2 (2x^3 + 1)^4 dx = ∫ (u^4) (du/6) = (1/6) ∫ u^4 du.

Integrate: ∫ u^4 du = u^5/5, so (1/6) ∫ u^4 du = (1/6) * (u^5/5) = u^5/30.

Substitute back: u = 2x^3 + 1, so the antiderivative is (2x^3 + 1)^5 / 30 + C.

If there were multiple options, you would compare each option to this result. Any option equivalent to (2x^3 + 1)^5 / 30 + C would be correct; any other form differing by a constant would also be correct, since they would represent the same family of antiderivatives.

Find the integral ∫ 𝑥 2 ( 2 𝑥 3 + 1 ) 4 𝑑 𝑥

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