Use a change of variables to evaluate the following integral. What is the best choice of u for the change of variables? Find du. \( x \sqrt{x^{2}-1600} \) du \( =\left(\frac{1}{2}\right) \) dx Rewrite the given integral using this change of variables. 41 \( \int_{40}^{4} x \sqrt{x^{2}-1,600} \) dx \( =\int(\sqrt{x}) \) du 40 (Type exact answers.) Evaluate the integral.

To evaluate the integral \( \int_{40}^{4} x \sqrt{x^{2}-1600} \, dx \) effectively, a good choice for the change of variable is \( u = x^2 - 1600 \). This choice simplifies the square root and directly relates \( dx \) to \( du \). Now, differentiating \( u \) gives us \( du = 2x \, dx \), so we can express \( dx \) as \( dx = \frac{du}{2x} \). You'll also need to adjust the limits of integration: when \( x = 40 \), \( u = 40^2 - 1600 = 0\), and when \( x = 4 \), \( u = 4^2 - 1600 = -1596 \). Rewriting the integral: \[ \int_{40}^{4} x \sqrt{x^{2}-1600} \, dx = \int_{0}^{-1596} \sqrt{u} \cdot \frac{du}{2} = \frac{1}{2} \int_{0}^{-1596} \sqrt{u} \, du \] Now evaluate the integral \( \int \sqrt{u} \, du = \frac{2}{3} u^{3/2} \), giving: \[ \frac{1}{2} \cdot \frac{2}{3} \left[ u^{3/2} \right]_{0}^{-1596} = \frac{1}{3} \left[ (-1596)^{3/2} - 0 \right] \] Calculating \( (-1596)^{3/2} \) involves some care because we are dealing with a negative number under the root. However, since the integral's limits imply continuity in real numbers, we focus on this change of variables approach to compute the integral while carefully noting that in certain contexts the integral may not yield a straightforward real number. Thus, the integral evaluates to a complex number for the current bounds, but if the intent was to find a conceptual approach, the steps here illustrate that journey!