The indefinite integral given by [math: ∫12xe3x2dx] is[Fill in the blank]

Question

BlackTom AI · Accepted Answer

Let u = 3x^2 so du/dx = 6x and du = 6x dx.

Relevant concepts: use u-substitution (the reverse of the chain rule) for integrals of the form ∫ f'(x)·g(f(x)) dx. Here the exponent 3x^2 has derivative proportional to x, so substitution simplifies the integral to a basic exponential.

Step-by-step calculation:
1. Start with the integral I = ∫ 12 x e^{3x^2} dx.
2. Factor the constant: I = 12 ∫ x e^{3x^2} dx.
3. Substitute u = 3x^2 ⇒ du = 6x dx ⇒ x dx = du/6.
4. Replace x dx and 3x^2: I = 12 ∫ e^{u} (du/6) = 12*(1/6) ∫ e^{u} du.
5. Simplify constant: 12*(1/6) = 2, so I = 2 ∫ e^{u} du = 2 e^{u} + C.
6. Back-substitute u = 3x^2: I = 2 e^{3x^2} + C.

Verification by differentiation: d/dx[2 e^{3x^2}] = 2 e^{3x^2} · (6x) = 12 x e^{3x^2}, which matches the integrand.

Why the correct answer is correct: The substitution reduced the integrand to the derivative of the exponential e^{u}, yielding the constant factor 2 after cancelation of 6 in du. Thus the antiderivative is 2 e^{3x^2} + C.

Why other options are incorrect (briefly):
- a. 12x e^{3x^2} + C: differentiating gives extra terms; not the antiderivative (would produce (12 + 72x^2)e^{3x^2}).
- c. e^{x^3} + C: exponent and derivative do not match the integrand.
- d. 2x e^{3x} + C, e. 3x e^{3x} + C, f. (1/3)e^{3x} + C: these have exponent 3x (not 3x^2) or incorrect coefficients; differentiation does not yield 12x e^{3x^2}.

Conclusion: The integral evaluates to 2 e^{3x^2} + C, matching answer b.

The indefinite integral given by [math: ∫12xe3x2dx] is[Fill in the blank]单项选择题

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