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

## Summary/Background

Substitution involves a few standard steps:

"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

- 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

"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

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## 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

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AP Calculus AB (USA) | 4 | Integration | Substitution method | - |

AP Calculus BC (USA) | 4 | Integration | Substitution method | - |

AQA A-Level (UK - Pre-2017) | C3 | Integration | Substitution method | - |

AQA A2 Maths 2017 | Pure Maths | Integration | Integration by Substitution | - |

AQA AS/A2 Maths 2017 | Pure Maths | Integration | Integration by Substitution | - |

CBSE XII (India) | Calculus | Integrals | Integration by substitution | - |

CCEA A-Level (NI) | C4 | Integration | Substitution method | - |

CIE A-Level (UK) | P3 | Integration | Substitution method | - |

Edexcel A-Level (UK - Pre-2017) | C4 | Integration | Substitution method | - |

Edexcel A2 Maths 2017 | Pure Maths | Integration | Integration by Substitution | - |

Edexcel AS/A2 Maths 2017 | Pure Maths | Integration | Integration by Substitution | - |

I.B. Higher Level | 6 | Integration | Substitution method | - |

Methods (UK) | M9 | Integration | Substitution method | - |

OCR A-Level (UK - Pre-2017) | C4 | Integration | Substitution method | - |

OCR A2 Maths 2017 | Pure Maths | Integration Techniques | Integration by Substitution | - |

OCR MEI A2 Maths 2017 | Pure Maths | Integration Techniques | Integration by Substitution | - |

OCR-MEI A-Level (UK - Pre-2017) | C3 | Integration | Substitution method | - |

Pre-U A-Level (UK) | 5 | Integration | Substitution method | - |

Scottish Advanced Highers | M1 | Integration | Substitution method | - |

Scottish (Highers + Advanced) | AM1 | Integration | Substitution method | - |

Universal (all site questions) | I | Integration | Substitution method | - |

WJEC A-Level (Wales) | C4 | Integration | Substitution method | - |