Chapter 7, Section 2  

Integration by Parts

 

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  ∫ xex  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) (exdx),  ∫ (ex) (xdx),  ∫ (1) (xexdx),  ∫ (xex) (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 = ex dx is the most complicated portion of the integrand that fits a basic integration formula.

dv = ex dx   -->    v = ∫ dv = ∫ exdx = ex 

u = x   -->    du= dx

 Now, integration by parts produces the following.

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

∫ xex  dx = x ex - ∫ ex dx      Substitution

              = x ex ex + C     Integrate.   

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

 
     

Summary of Common Integrals Using Integration by Parts

1.  Integrals of  the form

∫ ( x n) ( e a xdx),  ∫ ( x n) (sin ax dx),  or  ∫ ( x n) (cos ax dx)

Let u =x n   

and let  dv = e a xdx, sin ax dx or cos ax dx.

2.  Integrals of  the form

∫ ( x n) (ln x dx),  ∫ ( x n) (arcsin ax dx),  or  ∫ ( x n) (arctan ax dx)

Let u = ln x, arcsin ax or arctan ax,  

and let  dv =xn dx

 

3.  Integrals of  the form

∫ ( e a x) (sin bx dx),   or  ∫ ( e a x) (cos ax dx)

Let u =  sin bx or  cos ax

and let  dv = e a x dx

 
 

TECHNIQUES OF INTEGRATION on S.O.S

Integration by Parts

Problems on Techniques of Integration

 

 
     

Link to Many Tables on S.O.S Math

 

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