1. **Problem Statement:** We are given the expression: $$\frac{x^{\left(-\frac{1}{2} + 1\right)}}{-\frac{1}{2} + 1} + \frac{x^{\left(\frac{1}{2} + 1\right)}}{\frac{1}{2} + 1} - \frac{2x^{0 + 1}}{0 + 1}$$ which simplifies to: $$\frac{x^{\frac{1}{2}}}{\frac{1}{2}} + \frac{x^{\frac{3}{2}}}{\frac{3}{2}} - 2x$$ 2. **Relevant Rules:** - Rule (1): $$\int f(x) f'(x) \, dx = \frac{(f(x))^{n+1}}{n+1}$$ - Rule (2): $$\int \frac{f'(x)}{f(x)} \, dx = \ln |f(x)|$$ - Substitution: Let $$u = f(x)$$ then $$\frac{du}{dx} = f'(x)$$ 3. **Step-by-step Explanation:** - The expression is a sum of terms of the form $$\frac{x^m}{m}$$ where $$m$$ is a power. - Recall that $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. - Here, the denominators correspond to the exponents plus one, matching the integral formula. 4. **Simplify each term:** - First term: $$\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}}$$ - Second term: $$\frac{x^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3}x^{\frac{3}{2}}$$ - Third term: $$-2x$$ remains as is. 5. **Final expression:** $$2x^{\frac{1}{2}} + \frac{2}{3}x^{\frac{3}{2}} - 2x$$ This expression represents the evaluated integral or simplified form based on the given rules. **Summary:** We used the power rule for integration and substitution to simplify the given expression step-by-step.