The integral represented by [math: ∫2xcos⁡(x2)−6x2sin⁡(x3)dx] is:[Fill in the blank]

Question

BlackTom AI · Accepted Answer

Use u-substitution: for 2x cos(x^2) take u = x^2, and for −6x^2 sin(x^3) take v = x^3.

1) Relevant knowledge points and concepts
- Basic antiderivative rules: ∫cos(u) du = sin(u), ∫sin(u) du = −cos(u).
- The u-substitution method: if u = g(x), then ∫f(g(x)) g'(x) dx = ∫f(u) du.
- Linear combination: integrate each term separately and combine with a constant of integration.

2) Complete reasoning process step by step
Split the integral:
∫(2x cos(x^2) − 6x^2 sin(x^3)) dx = ∫2x cos(x^2) dx − ∫6x^2 sin(x^3) dx.

First term: let u = x^2. Then du = 2x dx, so 2x dx = du. Thus
∫2x cos(x^2) dx = ∫cos(u) du = sin(u) + C1 = sin(x^2) + C1.

Second term: let v = x^3. Then dv = 3x^2 dx, so x^2 dx = dv/3. Thus
∫6x^2 sin(x^3) dx = 6 ∫x^2 sin(v) dx = 6 ∫sin(v) (dv/3) = (6/3) ∫sin(v) dv = 2 ∫sin(v) dv.
Then ∫sin(v) dv = −cos(v) + C2, so 2 ∫sin(v) dv = 2(−cos(v)) + C2' = −2 cos(v) + C2'.
Because the original integrand has a minus sign for this term, we had −∫6x^2 sin(x^3) dx = −(6-term) = −(2 ∫sin v dv) = −(−2 cos v) = +2 cos(v).
Equivalently more directly:
−∫6x^2 sin(x^3) dx = −6 * (1/3) * (−cos(x^3)) = +2 cos(x^3).

3) Intermediate calculation summary
- ∫2x cos(x^2) dx = sin(x^2).
- −∫6x^2 sin(x^3) dx = +2 cos(x^3).
Combine: sin(x^2) + 2 cos(x^3) + C.

4) Why the correct answer is correct
Each part matches a standard u-substitution where the derivative of the inner function (x^2 or x^3) appears as a factor (2x or 3x^2). The algebra of constants (6 * 1/3 = 2 and the extra minus sign) produces +2 cos(x^3). Therefore the combined antiderivative is sin(x^2) + 2 cos(x^3) + C.

5) Why other options are incorrect (brief)
- a: has −sin(x^2) and a 2x cos(x^3) term — signs and factors are wrong and 2x cos(x^3) is not the correct antiderivative of −6x^2 sin(x^3).
- b: uses x cos(...) forms that don't arise from correct substitution.
- c: cos(x2)+cos(x) is incorrect inner arguments and missing correct coefficients.
- e and f: include polynomial factors multiplied by trig terms (like x^2 cos(x^2) or x^3 sin(x^3)), which are not the results of standard substitution here.

6) Conclusion
The antiderivative is sin(x^2) + 2 cos(x^3) + C, which matches the given correct choice d.

The integral represented by [math: ∫2xcos⁡(x2)−6x2sin⁡(x3)dx] is:[Fill in the blank]单项选择题

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