Calculate

Question

Calculate

BlackTom AI · Accepted Answer

Let u = x^2, which directly handles the factor x e^{x^2+2}.

1) Relevant concepts:
- Exponential rules: e^{x^2+2} = e^{2} e^{x^2}, so e^{2} is a constant factor.
- Substitution for integrals: if u = x^2 then du = 2x dx, so x dx = du/2.
- Basic antiderivative: ∫ e^{u} du = e^{u} + C.

2) Step-by-step reasoning and intermediate steps:
Start with the integral I = ∫_{0}^{√(ln 2)} x e^{x^2+2} dx.
Factor out e^{2} since it does not depend on x:
I = e^{2} ∫_{0}^{√(ln 2)} x e^{x^2} dx.

Make substitution u = x^2. Then du = 2x dx ⇒ x dx = du/2.
Change limits: when x = 0, u = 0^2 = 0. When x = √(ln 2), u = (√(ln 2))^2 = ln 2.

Thus
I = e^{2} ∫_{u=0}^{u=ln 2} (1/2) e^{u} du
  = (e^{2}/2) ∫_{0}^{ln 2} e^{u} du.

Compute the antiderivative:
∫_{0}^{ln 2} e^{u} du = [e^{u}]_{0}^{ln 2} = e^{ln 2} - e^{0} = 2 - 1 = 1.

So I = (e^{2}/2) · 1 = e^{2}/2.

3) Why the correct answer is correct:
The substitution reduced the integrand to a simple exponential whose integral is itself; factoring e^{2} made the constant multiplier explicit, and the definite limits converted to 0 and ln 2 give the final value 1 for the inner integral. Multiplying by e^{2}/2 yields e^{2}/2.

4) Why other options are incorrect (brief):
- a (e^2): misses the 1/2 factor and the subtraction from the limits.
- b (1/2), c (1), d (e), e (2), g (ln 2), h (0): all ignore one or more factors (either missing e^{2} factor or 1/2 from substitution or incorrect evaluation of the exponential limits). None match the exact algebraic result e^{2}/2.

Final confirmation: the evaluated integral equals e^{2}/2, which corresponds to option f.

Calculate单项选择题

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