FAQ Topic: 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

$latex \int{u{v}’ dx=uv-\int{v{u}’ dx}}$

Below are several examples that students worked out.

Problem 1 –

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

Problem 2 –

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

Problem 3 –

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

 

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

Example 1

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 2

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

 

 

 

 

 

Example 3

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

 

 

 

 

 

Example 4

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

 

 

 

 

Example 5

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 6

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

 

 

 

 

 Example 7

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