Integration with Chain Rule

Introduction

Integrating with the chain rule is the opposite of taking the derivative with the chain rule. For the derivative of the chain rule, picture.
So picture 

For example:

1.1   picture 
       =picture

1.2   picture 
       picture

1.3   picture 
       picture

1.4   picture 
       picture

1.5   picture 
       picture 
       picture

The same holds true for trigonometric functions too.
picture 

For example:

2.1   picture 
       picture

2.2   picture 
       picture

2.3   picture 
       picture 
       picture 
       picture

2.4   picture 
       picture

2.5   picture
       picture 
       picture

2.6   picture
       picture

2.7   picture
       picture 
       picture

2.8   picture
       picture