1. **Problem:** Find the integral $$\int \frac{10}{x^{5/2} \sqrt{x^2}} \, dx$$. 2. **Recall:** $$\sqrt{x^2} = |x|$$, but for integration purposes and assuming $$x > 0$$, $$\sqrt{x^2} = x$$. 3. **Simplify the integrand:** $$\frac{10}{x^{5/2} \cdot x} = \frac{10}{x^{5/2 + 1}} = \frac{10}{x^{7/2}} = 10x^{-7/2}$$. 4. **Use the power rule for integration:** $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$. 5. **Apply the rule:** $$\int 10x^{-7/2} \, dx = 10 \int x^{-7/2} \, dx = 10 \cdot \frac{x^{-7/2 + 1}}{-7/2 + 1} + C = 10 \cdot \frac{x^{-5/2}}{-5/2} + C$$. 6. **Simplify the fraction:** $$10 \cdot \frac{x^{-5/2}}{-5/2} = 10 \cdot \left(-\frac{2}{5}\right) x^{-5/2} = -\frac{20}{5} x^{-5/2} = -4 x^{-5/2} + C$$. 7. **Rewrite with radicals:** $$x^{-5/2} = \frac{1}{x^{5/2}} = \frac{1}{x^2 \sqrt{x}}$$. 8. **Final answer:** $$-\frac{4}{x^2 \sqrt{x}} + C$$. **Check options:** None exactly match this form, but option A is closest in structure with a negative sign and denominator involving $$\sqrt{x^2}$$. Given the problem's options, the closest correct answer is A. **Answer:** A. - (25 / (5\sqrt{x^2})) + C (Note: The problem's options seem inconsistent with the integral's exact evaluation, but A is the best match.)