Integration & Antiderivatives

Emily Davis

7 min read

Study Guide Overview

This study guide covers derivatives and antiderivatives (indefinite integrals). It explores the indefinite integrals of common functions including powers of x, exponentials, logarithmic functions, trigonometric functions, reciprocal trigonometric functions, and inverse trigonometric functions. A table of common indefinite integrals is provided for easy reference, along with practice questions and a glossary of key terms.

#Derivatives & Antiderivatives

#Introduction

Understanding derivatives and antiderivatives (indefinite integrals) is crucial in calculus. Differentiation and integration are inverse operations, meaning you can often reverse the process of differentiation to find the indefinite integrals of functions.

Exam Tip

Link to IB Curriculum: This section covers the fundamental concepts of calculus, focusing on differentiation and integration, which are essential for understanding the rate of change and the area under curves.

#Indefinite Integrals of Common Functions

#Indefinite Integrals of Powers of $x$

If you know the derivative of a function, you can find its indefinite integral. For any function $f(x)$ :

$f'(x) = g(x) \implies \int g(x) , dx = f(x) + C$

Key Concept

Key Concept: If $\frac{d}{dx}(x^n) = nx^{n-1}$ , then $\int x^n , dx = \frac{1}{n+1} x^{n+1} + C$ for $n \neq -1$ .

Special Cases:

$\int k , dx = kx + C$ , where $k$ is a constant
$\int 0 , dx = C$

**Worked Example:**

Find the indefinite integral $\int x^{10} , dx$ .

Solution:

Using $\int x^n , dx = \frac{1}{n+1} x^{n+1} + C$ ,

$\int x^{10} , dx = \frac{1}{10+1} x^{10+1} + C = \frac{1}{11} x^{11} + C$

#Indefinite Integrals of Exponentials and $\frac{1}{x}$

For exponential functions:

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Integration & Antiderivatives

Emily Davis

7 min read

Next Topic - Indefinite Integral Rules

Study Guide Overview

#Derivatives & Antiderivatives

#Table of Contents

Introduction
Indefinite Integrals of Common Functions
Table of Common Indefinite Integrals
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction

Exam Tip

#Indefinite Integrals of Common Functions

#Indefinite Integrals of Powers of $x$

If you know the derivative of a function, you can find its indefinite integral. For any function $f(x)$ :

$f'(x) = g(x) \implies \int g(x) , dx = f(x) + C$

Key Concept

Key Concept: If $\frac{d}{dx}(x^n) = nx^{n-1}$ , then $\int x^n , dx = \frac{1}{n+1} x^{n+1} + C$ for $n \neq -1$ .

Special Cases:

$\int k , dx = kx + C$ , where $k$ is a constant
$\int 0 , dx = C$

**Worked Example:**

Find the indefinite integral $\int x^{10} , dx$ .

Solution:

Using $\int x^n , dx = \frac{1}{n+1} x^{n+1} + C$ ,

$\int x^{10} , dx = \frac{1}{10+1} x^{10+1} + C = \frac{1}{11} x^{11} + C$

#Indefinite Integrals of Exponentials and $\frac{1}{x}$

For exponential functions:

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Previous Topic - Indefinite Integrals Next Topic - Indefinite Integral Rules

Integration & Antiderivatives

Emily Davis

#Derivatives & Antiderivatives

#Table of Contents

#Introduction

#Indefinite Integrals of Common Functions

#Indefinite Integrals of Powers of xxx

#Indefinite Integrals of Exponentials and 1x\frac{1}{x}x1​

How are we doing?

Integration & Antiderivatives

Emily Davis

#Derivatives & Antiderivatives

#Table of Contents

#Introduction

#Indefinite Integrals of Common Functions

#Indefinite Integrals of Powers of xxx

#Indefinite Integrals of Exponentials and 1x\frac{1}{x}x1​

How are we doing?

#Indefinite Integrals of Powers of $x$

#Indefinite Integrals of Exponentials and $\frac{1}{x}$

#Indefinite Integrals of Powers of $x$

#Indefinite Integrals of Exponentials and $\frac{1}{x}$