Go to content

Integrate \displaystyle \int \frac{1}{x}[1+(\ln x)^2]\, dx

## Summary/Background

Substitution involves a few standard steps:
• Choose a suitable substitution, say u = f(x)
• Differentiate the substitution, to get \frac{du}{dx} = f'(x)
• Rearrange this derivative to get du = f'(x)dx
• Substitute both u = f(x) and du = f'(x)dx into the integral and see if you can simplify the result
Normally the necessary substitution will be given on the exam paper. Many functions that can be integrated by the substitution method can also be integrated by "inspection" or "observation". This means that they are in a form that fits a pattern. There are two common types:

"Top is derivative of bottom": \displaystyle \int \frac{f'(x)}{f(x)}dx = \ln[f(x)] + c

"A function times its derivative": \displaystyle \int f^n (x)f'(x)dx = \frac{1}{n+1}f^{n+1}(x) + c

This page displays a video produced by MidnightTutor.com and uploaded to YouTube. Note that many school and college computer networks do not allow access to YouTube videos, so you may have to view this page from a stand-alone computer.

## Glossary

### derivative

rate of change, dy/dx, f'(x), , Dx.

### differentiate

to find the derivative of a function

### function

A rule that connects one value in one set with one and only one value in another set.

### integral

the anti-derivative

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus AB (USA)4IntegrationSubstitution method-
AP Calculus BC (USA)4IntegrationSubstitution method-
AQA A-Level (UK - Pre-2017)C3IntegrationSubstitution method-
AQA A2 Maths 2017Pure MathsIntegrationIntegration by Substitution-
AQA AS/A2 Maths 2017Pure MathsIntegrationIntegration by Substitution-
CBSE XII (India)CalculusIntegralsIntegration by substitution-
CCEA A-Level (NI)C4IntegrationSubstitution method-
CIE A-Level (UK)P3IntegrationSubstitution method-
Edexcel A-Level (UK - Pre-2017)C4IntegrationSubstitution method-
Edexcel A2 Maths 2017Pure MathsIntegrationIntegration by Substitution-
Edexcel AS/A2 Maths 2017Pure MathsIntegrationIntegration by Substitution-
I.B. Higher Level6IntegrationSubstitution method-
Methods (UK)M9IntegrationSubstitution method-
OCR A-Level (UK - Pre-2017)C4IntegrationSubstitution method-
OCR A2 Maths 2017Pure MathsIntegration TechniquesIntegration by Substitution-
OCR MEI A2 Maths 2017Pure MathsIntegration TechniquesIntegration by Substitution-
OCR-MEI A-Level (UK - Pre-2017)C3IntegrationSubstitution method-
Pre-U A-Level (UK)5IntegrationSubstitution method-
Scottish Advanced HighersM1IntegrationSubstitution method-
Scottish (Highers + Advanced)AM1IntegrationSubstitution method-
Universal (all site questions)IIntegrationSubstitution method-
WJEC A-Level (Wales)C4IntegrationSubstitution method-