\( \begin{array}{l} \text { Use a change of variables to evaluate the following definite integral. }\\ \int_{0}^{1} \frac{2 x}{\left(x^{2}+5\right)^{3}} d x \end{array} \)

To evaluate the integral \( \int_{0}^{1} \frac{2 x}{(x^{2}+5)^{3}} \, dx \), we can use the substitution \( u = x^2 + 5 \). Thus, we have: 1. The differential change is given by \( du = 2x \, dx \), which means \( dx = \frac{du}{2x} \). Now substitute \( x = 0 \) and \( x = 1 \): - When \( x = 0 \), \( u = 0^2 + 5 = 5 \). - When \( x = 1 \), \( u = 1^2 + 5 = 6 \). So the limits of integration change from 0 and 1 to 5 and 6, respectively. The integral becomes: \[ \int_{5}^{6} \frac{1}{u^{3}} \, du \] 2. Now, we'll compute this integral: \[ \int \frac{1}{u^{3}} \, du = -\frac{1}{2u^{2}} + C \] Evaluating this from 5 to 6 gives us: \[ \left[-\frac{1}{2u^{2}}\right]_{5}^{6} = -\frac{1}{2(6)^{2}} + \frac{1}{2(5)^{2}} = -\frac{1}{72} + \frac{1}{50} \] To combine these fractions, find a common denominator, which is 360: \[ -\frac{1}{72} = -\frac{5}{360}, \quad \frac{1}{50} = \frac{7.2}{360} \] Now, we can subtract: \[ \frac{7.2}{360} - \frac{5}{360} = \frac{2.2}{360} = \frac{11}{1800} \] Therefore, the value of the definite integral \( \int_{0}^{1} \frac{2 x}{(x^{2}+5)^{3}} \, dx \) is: \[ \frac{11}{1800} \]