Question

use the substitution formula, (int_{a}^{b} f(g(x)) cdot g(x) , dx = int_{g(a)}^{g(b)} f(u) , du) where (g(x) = u), to evaluate the following integral (int_{1}^{3} \frac{4(ln x)^3}{x} , dx). determine a change of variables from (y) to (u). choose the correct answer below: (\bigcirc) a. (u = x) (\bigcirc) b. (u = \frac{4(ln x)^3}{x}) (\bigcirc) c. (u = 4(ln x)^3) (\bigcirc) d. (u = ln x)

Explanation:

To determine the change of variables \( u \) for the integral \( \int_{1}^{3} \frac{4(\ln x)^{3}}{x} dx \) using substitution, we look for a function \( g(x) \) such that its derivative \( g'(x) \) is also present in the integrand.

Let's analyze each option:

• Option A: \( u = x \). Then \( du = dx \), but the integrand has \( \frac{(\ln x)^{3}}{x} \), and \( du = dx \) doesn't match the \( \frac{1}{x} dx \) part. So this is incorrect.
• Option B: \( u=\frac{4(\ln x)^{3}}{x} \). The derivative of this \( u \) would be complicated and not match the simple structure of the integrand. So this is incorrect.
• Option C: \( u = 4(\ln x)^{3} \). The derivative \( du=4\times3(\ln x)^{2}\times\frac{1}{x}dx = 12\frac{(\ln x)^{2}}{x}dx \), which is not the same as the \( \frac{4(\ln x)^{3}}{x}dx \) or its parts. So this is incorrect.
• Option D: \( u=\ln x \). Then \( du=\frac{1}{x}dx \), which is exactly the \( \frac{1}{x}dx \) part in the integrand \( \frac{4(\ln x)^{3}}{x}dx = 4(\ln x)^{3}\times\frac{1}{x}dx=4u^{3}du \) (after substitution). So this is the correct choice.

Answer:

D. \( u = \ln x \)