Indefinite Integrals of Sums, Differences, and Constant Multiples

Table of Contents

1. Introduction
2. Integrating Sums and Differences
3. Integrating Constant Multiples
4. Combining Rules
5. Simplifying Expressions
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

---

Introduction

In calculus, integrating sums, differences, and constant multiples of functions is a fundamental skill. This guide will help you understand how to handle these types of integrals, providing rules, examples, and practice questions.

---

Integrating Sums and Differences

Key Concept

When integrating sums or differences of functions, the integral is simply the sum or difference of the integrals of the individual functions.

$$\int \left( f(x) \pm g(x) \right) , dx = \int f(x) , dx \pm \int g(x) , dx$$

**Example:** $$\int \left( x^2 + \cos x \right) \, dx = \int x^2 \, dx + \int \cos x \, dx = \frac{1}{3} x^3 + \sin x + C$$

Note: We still only need a single constant of integration, $C$.

---

Integrating Constant Multiples

Key Concept

When integrating a constant multiple of a function, the constant can be pulled out in front of the integral.

$$\int k f(x) , dx = k \int f(x) , dx$$

**Example:** $$\int 5 e^x \, dx = 5 \int e^x \, dx = 5 e^x + C$$

---

Combining Rules

Key Concept

The rules for sums, differences, and constant multiples can be combined. If $p$ and $q$ are constants, then:

$$\int \left( p f(x) \pm q g(x) \right) , dx = p \int f(x) , dx \pm q \int g(x) , dx$$

This idea can be extended for any number of terms inside the integral.

---

Worked Example

Problem

Find the indefinite integral:

$$\int \left( 3x^3 - 2 \sin 3x + 5 e^{4x} \right) , dx$$

Solution

1. Write as a sum or difference of integrals, with constants pulled in front as multipliers:

   $$3 \int x^3 , dx - 2 \int \sin 3x , dx + 5 \int e^{4x} , dx$$

2. Integrate each term:

   $$3 \cdot \frac{1}{4} x^4 - 2 \cdot \left( -\frac{1}{3} \cos 3x \right) + 5 \cdot \frac{1}{4} e^{4x} + C$$

3. Simplify:

   $$\frac{3}{4} x^4 + \frac{2}{3} \cos 3x + \frac{5}{4} e^{4x} + C$$

---

Simplifying Expressions

Key Concept

Sometimes expressions need to be simplified before integration. This may involve expanding brackets or rearranging fractions.

Example 1: Expanding Brackets

Problem: $$\int \left( x^2 + 2 \right)^2 , dx$$

Solution:

1. Expand the brackets:

   $$\left( x^2 + 2 \right)^2 = x^4 + 4x^2 + 4$$

2. Integrate each term:

   $$\int \left( x^4 + 4x^2 + 4 \right) , dx = \frac{1}{5} x^5 + \frac{4}{3} x^3 + 4x + C$$

Example 2: Rearranging Fractions

Problem: $$\int \frac{5x^3 - 3}{x^2} , dx$$

Solution:

1. Rearrange the fraction:

   $$\frac{5x^3 - 3}{x^2} = \frac{5x^3}{x^2} - \frac{3}{x^2} = 5x - \frac{3}{x^2}$$

2. Use laws of exponents:

   $$5x - \frac{3}{x^2} = 5x - 3x^{-2}$$

3. Integrate each term:

   $$\int \left( 5x - 3x^{-2} \right) , dx = \frac{5}{2} x^2 + 3x^{-1} + C = \frac{5}{2} x^2 + \frac{3}{x} + C$$

---

Practice Questions

Practice Question

Question 1: Integrate the following expression: $$\int (4x^2 - 7 \cos x) , dx$$

Answer: $$\frac{4}{3} x^3 - 7 \sin x + C$$

Practice Question

Question 2: Integrate the following expression: $$\int 2e^{2x} + 3x^3 , dx$$

Answer: $$e^{2x} + \frac{3}{4} x^4 + C$$

Practice Question

Question 3: Simplify and integrate: $$\int \frac{6x^2 + 8}{x} , dx$$

Answer: $$\int (6x + \frac{8}{x}) , dx = 3x^2 + 8 \ln |x| + C$$

---

Glossary

• Integral: The anti-derivative of a function, representing the area under the curve.
• Constant of Integration (C): An arbitrary constant added to the indefinite integral of a function.
• Exponent: The power to which a number or expression is raised.

---

Summary and Key Takeaways

Summary

• Sums and Differences: Integrate each term individually.
• Constant Multiples: Pull the constant out in front of the integral.
• Combining Rules: Combine the rules for sums, differences, and constant multiples.
• Simplifying Expressions: Expand or rearrange expressions before integrating if necessary.

Key Takeaways

1. The integral of a sum or difference of functions is the sum or difference of their integrals.
2. Constants can be factored out of integrals.
3. Simplify expressions before integrating for easier computation.

---

**Note:** For complex integrals, consider breaking them down into simpler parts or using advanced techniques like integration by parts or substitution.

---

By following these guidelines and practicing regularly, you'll be well-prepared to tackle integrals involving sums, differences, and constant multiples confidently.