What is Integration by Parts?

This process for reversing the Product Rule for Derivatives is called Integration by Parts . It is covered in Section 14.2. In Integration by Parts, the integrand (the thing you are finding the antiderivative of) is written as a product. One piece is thought of as u and the other part v‘. The formula then says

\int{u{v}' dx=uv-\int{v{u}' dx}}

Below are several examples that students worked out.

Problem 1

\displaystyle \int{\left( 1-x \right){{e}^{x}} dx}

Problem 2

\displaystyle \int {\left( 8x+10 \right) \ln \left( x \right) dx}

Problem 3

\displaystyle \int{\left( 2t-1 \right) \ln \left( t \right) dt}

 

Do You Have More Integration By Parts Examples?

Here are some more examples of integration by parts courtesy of the face-to-face class.

Example 1

displaystyle int{left( 2t-1 right)ln left( t right)dt}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 2

displaystyle int{{{x}^{4}}ln left( x right)dx}

 

 

 

 

 

Example 3

displaystyle int{ln left( 2x right)dx}

 

 

 

 

 

Example 4

displaystyle int{left( 5z-4 right)ln left( z right)dz}

 

 

 

 

Example 5

displaystyle int{{{x}^{3}}ln left( x right)dx}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 6

displaystyle int{{{x}^{2}}ln left( x right)dx}

 

 

 

 

 Example 7

displaystyle int{ln left( 3z right)dz}