Methods of Integration

Michael Green
8 min read
Listen to this study note
Study Guide Overview
This guide covers integration by substitution (u-substitution), including the basic steps, working with indefinite and definite integrals, and handling more complex substitutions. It provides worked examples, practice questions, and a glossary of key terms. Key takeaways and common mistakes are also highlighted.
Table of Contents
- Introduction
- Integration by Substitution
- Advanced -Substitution
- Definite Integrals Using -Substitution
- Practice Questions
- Glossary
- Summary and Key Takeaways
Introduction
In calculus, integration by substitution (also known as -substitution) is a method used to find integrals. It simplifies an integral by transforming it into an easier one through a change of variable.
Integration by Substitution
Basic Steps for -Substitution
What is integration by substitution?
Substitution simplifies an integral by defining an alternative variable (usually ) in terms of the original variable (usually ). The new integral in is often easier to solve, and the substitution can be reversed at the end to express the answer in terms of .
How do I integrate simple functions using -substitution?
In a simple integral involving substitution, you will usually be integrating a composite function (i.e., 'function of a function'). Substitution can be a safer method when integrating by inspection is awkward or difficult to spot.
Steps for -Substitution:
-
Identify the substitution to be used – it will be the secondary (or 'inside') function in a composite function.
- If the integral involves , let .
- Example:
- Let .
-
Differentiate the substitution and rearrange.
- can be treated like a fraction. Multiply by to get rid of fractions.
- Example:
- Then .
-
Replace all parts of the integral.
- Replace all terms, including .
- Example:
- So .
-
Integrate.
- Example:
- Don't forget the constant of integration.
- Example:
-
Substitute back in.
- Replace everywhere with the equivalent expression for .
- Example:
- So .
Worked Example 1
Find the indefinite integral .
Solution:
-
Identify the substitution:
- Let .
-
Differentiate and rearrange:
- .
-
Replace all parts of the integral:
- .
-
Integrate:
- .
-
Substitute back in:
- .
Advanced -Substitution
The procedure here is exactly the same as for integrating simpler functions, but the substitution to use may not be as obvious.
How do I integrate more complicated functions using -substitution?
- Practice more complex questions to improve your integration by substitution skills.
- Example:
- Identify the substitution: Let .
- Differentiate and rearrange: .
- Replace all parts of the integral: .
- So .
- Integrate: .
- Substitute back in: .
- Example:
Worked Example 2
Find the indefinite integral .
Solution:
-
Identify the substitution:
- Recall the standard integral .
- Rearrange the integral: .
-
Substitute:
- Let .
-
Differentiate and rearrange:
- .
-
Replace all parts of the integral:
- .
-
Integrate:
- .
-
Substitute back in:
- .
Definite Integrals Using -Substitution
How do I evaluate definite integrals using -substitution?
Definite integrals can also be solved using -substitution. You just need to rewrite the integration limits in terms of as well.
Example:
-
Substitution:
- Let .
-
Differentiate and rearrange:
- .
-
Change the integral limits:
- When , .
- When , .
-
Replace all parts of the integral:
- .
-
Integrate and evaluate the definite integral using the values:
- .
Worked Example 3
Evaluate the definite integral .
Solution:
-
Identify the substitution:
- Let .
-
Differentiate and rearrange:
- .
-
Find the integration limits in terms of :
- When , .
- When , .
-
Replace all parts of the integral, including the integration limits:
- .
-
Integrate and evaluate the definite integral:
- .
-
Use laws of logarithms to simplify the final answer:
- .
Practice Questions
Practice Question
-
Find the indefinite integral .
-
Evaluate the definite integral .
-
Find the indefinite integral .
-
Evaluate the definite integral .
-
Find the indefinite integral .
Glossary
- Composite Function: A function composed of two functions such that the output of one function becomes the input of the other.
- Constant of Integration: An arbitrary constant added to the function after integration, denoted by .
- Definite Integral: An integral with upper and lower limits, which gives the area under the curve between those limits.
- Indefinite Integral: An integral without limits, representing a family of functions.
- Substitution: A method used to simplify an integral by introducing a new variable.
Summary and Key Takeaways
Key Points:
- Substitution is a powerful method for simplifying integrals, especially for composite functions.
- Identify the inner function in a composite function to make the correct substitution.
- Differentiate and rearrange the substitution to replace all parts of the integral.
- Integrate the simplified integral and always remember to substitute back the original variable.
Key Takeaways:
- Practice identifying appropriate substitutions in various integrals.
- Remember to change the limits of integration when evaluating definite integrals.
- Regular practice with -substitution will improve your problem-solving skills in calculus.
Exam Tips:
- Always double-check your substitution and differentiation steps.
- Ensure all parts of the integral are replaced correctly before integrating.
- Don't forget to add the constant of integration for indefinite integrals.
- For definite integrals, remember to adjust the integration limits according to the substitution.
Common Mistakes:
- Forgetting to change the limits of integration when evaluating definite integrals.
- Not substituting back the original variable after integrating.
- Incorrectly differentiating the substitution.
- Omitting the constant of integration for indefinite integrals.

How are we doing?
Give us your feedback and let us know how we can improve
Question 1 of 9
For the integral , what is the appropriate substitution and the resulting ?
,
,
,
,