Let's solve these integrals step by step for questions 11 to 14:

11. $\int \frac{1}{2}x^5 \, dx$

Solution:

1. Factor out $\frac{1}{2}$: $\int \frac{1}{2}x^5 \, dx = \frac{1}{2} \int x^5 \, dx$

2. Apply the power rule for integration: The rule states $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. $\frac{1}{2} \int x^5 \, dx = \frac{1}{2} \cdot \frac{x^{5+1}}{5+1} + C = \frac{1}{2} \cdot \frac{x^6}{6} + C$

3. Simplify: $\frac{x^6}{12} + C$

Answer: $\int \frac{1}{2}x^5 \, dx = \frac{x^6}{12} + C$

12. $\int (2 + 4x)^5 \, dx$

Solution:

1. Use substitution: Let $u = 2 + 4x$, so $du = 4 \, dx$ or $dx = \frac{du}{4}$.

2. Rewrite the integral in terms of $u$: $\int (2 + 4x)^5 \, dx = \int u^5 \cdot \frac{du}{4} = \frac{1}{4} \int u^5 \, du$

3. Integrate using the power rule: $\frac{1}{4} \int u^5 \, du = \frac{1}{4} \cdot \frac{u^{5+1}}{5+1} + C = \frac{1}{4} \cdot \frac{u^6}{6} + C = \frac{u^6}{24} + C$

4. Substitute back $u = 2 + 4x$: $\frac{(2 + 4x)^6}{24} + C$

Answer: $\int (2 + 4x)^5 \, dx = \frac{(2 + 4x)^6}{24} + C$

13. $\int (x + 5)^6 \, dx$

Solution:

1. Use substitution: Let $u = x + 5$, so $du = dx$.

2. Rewrite the integral in terms of $u$: $\int (x + 5)^6 \, dx = \int u^6 \, du$

3. Integrate using the power rule: $\int u^6 \, du = \frac{u^{6+1}}{6+1} + C = \frac{u^7}{7} + C$

4. Substitute back $u = x + 5$: $\frac{(x + 5)^7}{7} + C$

Answer: $\int (x + 5)^6 \, dx = \frac{(x + 5)^7}{7} + C$

14. $\int (3x + 1)^6 \, dx$

Solution:

1. Use substitution: Let $u = 3x + 1$, so $du = 3 \, dx$ or $dx = \frac{du}{3}$.

2. Rewrite the integral in terms of $u$: $\int (3x + 1)^6 \, dx = \int u^6 \cdot \frac{du}{3} = \frac{1}{3} \int u^6 \, du$

3. Integrate using the power rule: $\frac{1}{3} \int u^6 \, du = \frac{1}{3} \cdot \frac{u^{6+1}}{6+1} + C = \frac{1}{3} \cdot \frac{u^7}{7} + C = \frac{u^7}{21} + C$

4. Substitute back $u = 3x + 1$: $\frac{(3x + 1)^7}{21} + C$

Answer: $\int (3x + 1)^6 \, dx = \frac{(3x + 1)^7}{21} + C$

Do you have any questions or need further clarification?

Here are 5 related questions for practice:

1. What is $\int (5x - 2)^3 \, dx$?
2. How do you solve $\int 4x^3 \, dx$ using the power rule?
3. What substitution would you use for $\int (7x + 3)^5 \, dx$?
4. How do you compute definite integrals involving substitutions, e.g., $\int_0^1 (x + 1)^2 \, dx$?
5. Can you generalize the solution for $\int (ax + b)^n \, dx$?

Tip: Always double-check your substitution step to ensure the $dx$ term is properly replaced!