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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. \( y' = \sin(2t) + \cos\left(\frac{t}{2}\right) \)
\[
y = \int \left(\sin(2t) + \cos\left(\frac{t}{2}\right)\right) \, dt = -\frac{1}{2} \cos(2t) + 2\sin\left(\frac{t}{2}\right) + C
\]

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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!

Find all possible functions with the given derivative from the exercises in the image.

This page explains how to find all possible functions by integrating given derivatives. It includes exercises involving polynomial expressions, such as x^2, 3x^2 + 2x - 1, and trigonometric functions like sin(2t) and cos(t/2). The solutions utilize key integration rules and theorems.

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