# 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