These exercises involve finding the original function $y$ by integrating the given derivative $y'$. Let's go through each exercise, one by one, and solve them by integrating the provided $y'$.

Exercise 33:

a. $y' = x$

$y = \int x \, dx = \frac{x^2}{2} + C$ where $C$ is the constant of integration.

b. $y' = x^2$

$y = \int x^2 \, dx = \frac{x^3}{3} + C$

c. $y' = x^3$

$y = \int x^3 \, dx = \frac{x^4}{4} + C$

Exercise 34:

a. $y' = 2x$

$y = \int 2x \, dx = x^2 + C$

b. $y' = 2x - 1$

$y = \int (2x - 1) \, dx = x^2 - x + C$

c. $y' = 3x^2 + 2x - 1$

$y = \int (3x^2 + 2x - 1) \, dx = x^3 + x^2 - x + C$

Exercise 35:

a. $y' = -\frac{1}{x^2}$

$y = \int -\frac{1}{x^2} \, dx = \frac{1}{x} + C$

b. $y' = 1 - \frac{1}{x^2}$

$y = \int \left(1 - \frac{1}{x^2}\right) \, dx = x + \frac{1}{x} + C$

c. $y' = 5 + \frac{1}{x^2}$

$y = \int \left(5 + \frac{1}{x^2}\right) \, dx = 5x - \frac{1}{x} + C$

Exercise 36:

a. $y' = \frac{1}{2\sqrt{x}}$

$y = \int \frac{1}{2\sqrt{x}} \, dx = \sqrt{x} + C$

b. $y' = \frac{1}{\sqrt{x}}$

$y = \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C$

c. $y' = 4x - \frac{1}{\sqrt{x}}$

$y = \int \left(4x - \frac{1}{\sqrt{x}}\right) \, dx = 2x^2 - 2\sqrt{x} + C$

Exercise 37:

a. $y' = \sin(2t)$

$y = \int \sin(2t) \, dt = -\frac{1}{2} \cos(2t) + C$

b. $y' = \cos\left(\frac{t}{2}\right)$

$y = \int \cos\left(\frac{t}{2}\right) \, dt = 2\sin\left(\frac{t}{2}\right) + C$

c. $y' = \sin(2t) + \cos\left(\frac{t}{2}\right)$

$y = \int \left(\sin(2t) + \cos\left(\frac{t}{2}\right)\right) \, dt = -\frac{1}{2} \cos(2t) + 2\sin\left(\frac{t}{2}\right) + C$

Would you like more details on any specific exercise or the steps of integration? Here are 5 questions to explore further:

1. How can you find the constant $C$ in an integral if additional conditions are provided?
2. What are the integration techniques for complex derivatives like $\frac{1}{\sqrt{x}}$?
3. How can you apply integration to real-life problems involving motion or growth?
4. What is the significance of trigonometric integration in physics?
5. How does the chain rule relate to these types of differentiation problems?

Tip: When integrating, always remember to include the constant of integration $C$, as it represents a family of functions!