Integral Evaluation 1F1364

1. **State the problem:** We need to evaluate the integral $$\int (4x - 2x^{-\frac{1}{3}} + \frac{5}{x^2}) \, dx.$$ 2. **Recall the anti-differentiation formulas:** - For any power function $$x^n$$ where $$n \neq -1$$, the integral is $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$$ - For $$\frac{1}{x^2}$$, rewrite as $$x^{-2}$$ to apply the power rule. 3. **Apply the integral to each term separately:** $$\int 4x \, dx = 4 \int x^1 \, dx = 4 \cdot \frac{x^{1+1}}{1+1} = 4 \cdot \frac{x^2}{2} = 2x^2.$$ $$\int -2x^{-\frac{1}{3}} \, dx = -2 \int x^{-\frac{1}{3}} \, dx = -2 \cdot \frac{x^{-\frac{1}{3} + 1}}{-\frac{1}{3} + 1} = -2 \cdot \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = -2 \cdot \frac{3}{2} x^{\frac{2}{3}} = -3x^{\frac{2}{3}}.$$ $$\int \frac{5}{x^2} \, dx = 5 \int x^{-2} \, dx = 5 \cdot \frac{x^{-2+1}}{-2+1} = 5 \cdot \frac{x^{-1}}{-1} = -5x^{-1} = -\frac{5}{x}.$$ 4. **Combine all results and add the constant of integration:** $$\int (4x - 2x^{-\frac{1}{3}} + \frac{5}{x^2}) \, dx = 2x^2 - 3x^{\frac{2}{3}} - \frac{5}{x} + C.$$ **Final answer:** $$\boxed{2x^2 - 3x^{\frac{2}{3}} - \frac{5}{x} + C}.$$