Find

∫
2
𝑥
3
𝑑
𝑥

Question

Find

∫
2
𝑥
3
𝑑
𝑥

BlackTom AI · Accepted Answer

Starting with the given integral, we interpret the problem as evaluating ∫ 2x^3 dx.

A helpful rule is the power rule for integration: ∫ x^n dx = x^{n+1}/(n+1) + C, provided n ≠ -1. Here n = 3, so ∫ x^3 dx = x^4/4.

Because the integrand has a constant factor 2, we pull it out: ∫ 2x^3 dx = 2 ∫ x^3 dx.

Substituting the power rule result gives 2 × (x^4/4) + C = x^4/2 + C.

Now, looking at the provided option, the expression shown is x^4/2 + C, which matches the derived antiderivative exactly.

If a student were to misapply the rule and write x^3/(3) or x^4/3, that would be incorrect because the exponent increase and the denominator must reflect n+1 = 4 in the numerator and 4 in the denominator, then multiplied by the constant 2. Alternatively, forgetting the constant multiplier 2 and writing x^4/4 would also be a common mistake. In both cases, the result would not align with the correct antiderivative x^4/2 + C.

Therefore, the correct antiderivative is x^4/2 + C, as represented by the provided option.

Find ∫ 2 𝑥 3 𝑑 𝑥

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