Chapter 7, Section 2  

Theorem 7.1 Integration by Parts

If u and v are functions of x and have continuous derivatives, then

           ∫ u dv  =  uv - ∫ v du 

Guidelines for Integration by Parts

1. Try letting dv be the most complicated portion of the integrand that fits a basic integration rule.  Then u will be the remaining factor(s) of the integrand.

2.  Try letting u be the portion of the integrand whose derivative is a function simpler than u.  Then dv will be the remaining factors(s) of the integrand.     

Example 1  Integration by Parts

Evaluate  ∫ xe^x  dx.

Solution:  To apply integration by parts, you need to write the integral in the form of   ∫ u  du. 

There are several ways to do this.

∫ (x) (e^xdx),  ∫ (e^x) (xdx),  ∫ (1) (xe^xdx),  ∫ (xe^x) (dx)

The guidelines (above in Guidelines for Integration by Parts), suggest choosing the 1st potion because the derivative of u = x in simpler than x, and dv = e^x dx is the most complicated portion of the integrand that fits a basic integration formula.

dv = e^x dx   -->    v = ∫ dv = ∫ e^xdx = e^x 

u = x   -->    du= dx

 Now, integration by parts produces the following.

  ∫ u dv  =  uv - ∫ v du           Integration by parts formula.

∫ xe^x  dx = x e^x - ∫ e^x dx      Substitution

              = x e^x -  e^x + C     Integrate.   

.  To Check this, differentiate x e^x -  e^x + C   to see that you obtain the original integrand. 

Summary of Common Integrals Using Integration by Parts