Here are the solutions to the integration problems. 1. Step 1: Apply the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$, and $\int k \,dx = kx + C$. $$ \int (x^7 + 1) \,dx = \frac{x^{7+1}}{7+1} + x + C $$ Step 2: Simplify the expression. $$ = \frac{x^8}{8} + x + C $$ The student's answer is ✅ Correct. $$\boxed{\frac{x^8}{8} + x + C}$$ 2. Step 1: Rewrite the integrand using exponent rules. $\sqrt[3]{x^2} = x^{2/3}$, so $\frac{1}{\sqrt[3]{x^2}} = x^{-2/3}$. $$ \int \frac{dx}{\sqrt[3]{x^2}} = \int x^{-2/3} \,dx $$ Step 2: Apply the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$. Here $n = -\frac{2}{3}$. $$ = \frac{x^{-2/3 + 1}}{-2/3 + 1} + C = \frac{x^{1/3}}{1/3} + C $$ Step 3: Simplify the expression. $$ = 3x^{1/3} + C $$ The student's answer is ✅ Correct. $$\boxed{3x^{1/3} + C}$$ 3. Step 1: Apply the power rule for integration to each term. $$ \int (3x^3 + 2x^2 - x + 5) \,dx = 3\frac{x^{3+1}}{3+1} + 2\frac{x^{2+1}}{2+1} - \frac{x^{1+1}}{1+1} + 5x + C $$ Step 2: Simplify each term. $$ = \frac{3x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + 5x + C $$ The student's answer is ✅ Correct. $$\boxed{\frac{3x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + 5x + C}$$ 4. Step 1: Simplify the integrand by dividing each term in the numerator by $x^2$. $$ \int \frac{x^2 + 3x^3 - 4x}{x^2} \,dx = \int \left(\frac{x^2}{x^2} + \frac{3x^3}{x^2} - \frac{4x}{x^2}\right) \,dx $$ $$ = \int (1 + 3x - 4x^{-1}) \,dx $$ Step 2: Apply the power rule for integration and the rule for $\int \frac{1}{x} \,dx = \ln|x| + C$. $$ = x + 3\frac{x^{1+1}}{1+1} - 4\ln|x| + C $$ Step 3: Simplify the expression. $$ = x + \frac{3x^2}{2} - 4\ln|x| + C $$ The student's answer is ❌ Incorrect. The first term of the integrand was simplified incorrectly. $$\boxed{x + \frac{3x^2}{2} - 4\ln|x| + C}$$