Let's integrate the problems from 3 to 13 in detail. Below, I'll explain how to solve each integral step by step.

3. $\int (5 - x) \, dx$:

Separate terms: $\int 5 \, dx - \int x \, dx$ The integrals are: $5x - \frac{x^2}{2} + C$ Answer: $5x - \frac{x^2}{2} + C$.

4. $\int (a + bx^2) \, dx$:

Separate terms: $\int a \, dx + \int bx^2 \, dx$ The integrals are: $ax + \frac{bx^3}{3} + C$ Answer: $ax + \frac{bx^3}{3} + C$.

5. $\int (8x^4 - 5x^2 + 2) \, dx$:

Separate terms: $\int 8x^4 \, dx - \int 5x^2 \, dx + \int 2 \, dx$ The integrals are: $\frac{8x^5}{5} - \frac{5x^3}{3} + 2x + C$ Answer: $\frac{8x^5}{5} - \frac{5x^3}{3} + 2x + C$.

6. $\int \frac{7}{x^5} \, dx$:

Rewrite as: $\int 7x^{-5} \, dx$ The integral is: $7 \cdot \frac{x^{-4}}{-4} + C = -\frac{7}{4x^4} + C$ Answer: $-\frac{7}{4x^4} + C$.

7. $\int \left(1 - \frac{x^2}{2}\right) \, dx$:

Separate terms: $\int 1 \, dx - \int \frac{x^2}{2} \, dx$ The integrals are: $x - \frac{x^3}{6} + C$ Answer: $x - \frac{x^3}{6} + C$.

8. $\int (b + x)^3 \, dx$:

Expand $(b + x)^3$: $(b + x)^3 = b^3 + 3b^2x + 3bx^2 + x^3$ Integrate term by term: $\int b^3 \, dx + \int 3b^2x \, dx + \int 3bx^2 \, dx + \int x^3 \, dx$ Result: $b^3x + \frac{3b^2x^2}{2} + bx^3 + \frac{x^4}{4} + C$ Answer: $b^3x + \frac{3b^2x^2}{2} + bx^3 + \frac{x^4}{4} + C$.

9. $\int \frac{dy}{3y - 5}$:

This is a standard logarithmic integral: $\frac{1}{3} \ln|3y - 5| + C$ Answer: $\frac{\ln|3y - 5|}{3} + C$.

10. $\int (6x - 1)^2 \, dx$:

Expand $(6x - 1)^2$: $(6x - 1)^2 = 36x^2 - 12x + 1$ Integrate term by term: $\int 36x^2 \, dx - \int 12x \, dx + \int 1 \, dx$ Result: $12x^3 - 6x^2 + x + C$ Answer: $12x^3 - 6x^2 + x + C$.

11. $\int \left(\sqrt{y} + \frac{1}{\sqrt{y}}\right) \, dy$:

Rewrite terms: $\int y^{1/2} \, dy + \int y^{-1/2} \, dy$ The integrals are: $\frac{2y^{3/2}}{3} + 2y^{1/2} + C$ Answer: $\frac{2y^{3/2}}{3} + 2y^{1/2} + C$.

12. $\int \left(x^{5/2} - \frac{1}{x^{5/2}}\right) \, dx$:

Rewrite terms: $\int x^{5/2} \, dx - \int x^{-5/2} \, dx$ The integrals are: $\frac{2x^{7/2}}{7} - \frac{2x^{-3/2}}{3} + C$ Answer: $\frac{2x^{7/2}}{7} - \frac{2}{3x^{3/2}} + C$.

13. $\int (x^2 + 5)^2 \, dx$:

Expand $(x^2 + 5)^2$: $(x^2 + 5)^2 = x^4 + 10x^2 + 25$ Integrate term by term: $\int x^4 \, dx + \int 10x^2 \, dx + \int 25 \, dx$ Result: $\frac{x^5}{5} + \frac{10x^3}{3} + 25x + C$ Answer: $\frac{x^5}{5} + \frac{10x^3}{3} + 25x + C$.

Would you like any step clarified or need further details? 😊

5 Related Questions:

1. How do we determine when to expand expressions like $(b + x)^3$ during integration?
2. What is the difference between logarithmic integrals and power-rule-based integrals?
3. How do we integrate composite functions without expanding them?
4. What are the steps for handling fractional powers in integrals?
5. Can definite integrals be applied to these examples for further practice?

Tip: Always rewrite complex expressions (like $x^{5/2}$ or fractions) in simpler exponential form before integration. It simplifies calculations!