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

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

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

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