Use of integration by parts \displaystyle\int f(x)\,g'(x)\,dx=f(x)\,g(x) - \displaystyle \int f'(x)\,g(x)\,dxrequires a suitable choice for f(x) and g'(x).Choose from the options below, the choice of f(x) and g'(x) that will NOT help find the corresponding indefinite integral.单项选择题
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Question

Use of integration by parts \displaystyle\int f(x)\,g'(x)\,dx=f(x)\,g(x) - \displaystyle \int f'(x)\,g(x)\,dxrequires a suitable choice for f(x) and g'(x).Choose from the options below, the choice of f(x) and g'(x) that will NOT help find the corresponding indefinite integral.单项选择题

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BlackTom AI · Accepted Answer

The question asks to identify which choice of f(x) and g'(x) will NOT help in applying integration by parts to the integral ∫ (ln(x))^2 / x^3 dx.

In integration by parts, we want to split the integrand into f(x) g'(x) so that the derivative f'(x) is simpler (or at least reduces a problematic part) and the integral of g'(x) is manageable to obtain g(x). A natural approach for ∫ (ln x)^2 x^{-3} dx is to take u = (ln x)^2 and dv = x^{-3} dx. Then f(x) = (ln x)^2 and g'(x) = x^{-3} (so g(x) = ∫ x^{-3} dx = -1/2 x^{-2}). This setup tends to simplify the problem after applying the formula.

The given option is: f(x) = x^3 and g'(x) = (ln x)^2. This pairing is inappropriate for several reasons:
- f(x) = x^3 has derivative f'(x) = 3x^2, which does not simplify the logarithmic part (ln x)^2 or reduce the power of x in the denominator; it actually makes the algebra more cumbersome.
- Integrating g'(x) = (ln x)^2 would require computing ∫ (ln x)^2 dx, which does not yield a simpler or readily elementary antiderivative in the context of the original integrand, defeating the purpose of IBP here.
- The product f(x) g'(x) = x^3 (ln x)^2 would give the original integrand multiplied by x^3, not matching ∫ (ln x)^2 / x^3 dx at all, so the setup is misaligned with the target integral.

Thus, this pairing will not assist in finding the indefinite integral via integration by parts.

If we were to propose a correct pairing instead, we would choose f(x) = (ln x)^2 and g'(x) = x^{-3}, as explained earlier, or f(x) = ln x and g'(x) = (ln x) x^{-3}, depending on the chosen IBP route, to progressively reduce the complexity.