zuai-logo

Integration & Antiderivatives

Emily Davis

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

Table of Contents

  1. Introduction
  2. Indefinite Integrals of Common Functions
  3. Table of Common Indefinite Integrals
  4. Practice Questions
  5. Glossary
  6. Summary and Key Takeaways

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 xx

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

f′(x)=g(x)  ⟹  ∫g(x),dx=f(x)+Cf'(x) = g(x) \implies \int g(x) , dx = f(x) + C

Key Concept

Key Concept: If ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}, then ∫xn,dx=1n+1xn+1+C\int x^n , dx = \frac{1}{n+1} x^{n+1} + C for n≠−1n \neq -1.

Special Cases:

  • ∫k,dx=kx+C\int k , dx = kx + C, where kk is a constant
  • ∫0,dx=C\int 0 , dx = C
**Worked Example:**

Find the indefinite integral ∫x10,dx\int x^{10} , dx.

Solution:

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

∫x10,dx=110+1x10+1+C=111x11+C\int x^{10} , dx = \frac{1}{10+1} x^{10+1} + C = \frac{1}{11} x^{11} + C

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

For exponential functions:

Previous Topic - Indefinite IntegralsNext Topic - Indefinite Integral Rules