AP CALCULUS NOTES
SECTION 8.2 INTEGRATION BY PARTS
The u-substitution method of integration that we previously examined is based on the chain
rule for differentiation. Integration by parts, which we examine in this section, is a method
that is based on the product rule for differentiation.
A.) Integration by Parts: Let u and v represent differentiable functions of x.
d
uv  
dx
 u dv  uv   v du
Ex.1.) Find the

x sin x dx .
Note: To use integration by parts successfully, the choice of u and v must be made so that the
new integral is easier than the original. When the integrand consists of a product of two different
types of functions, let u designate the function that appears first in the acronym LIATE and let
dv denote the rest of the integrand.
Ex.2.)

x ln x dx
Repeated Integration by Parts:
Ex.3.)  x 2 e x dx
Ex.4.)
e
x
sin x dx
Performing Repeated Integration by Parts using Tabular Integration – one of the functions
in the integrand is a polynomial.
Steps for Tabular Integration:
1.) Repeatedly differentiate the polynomial (left column) until you get a 0 entry.
2.) Repeatedly antidifferentiate the other function (right column).
3.) Connect the columns with diagonal segments that alternate signs from + to – .
Ex.5.)  x3 sin x dx
C.) Integration by Parts for Definite Integrals:
Ex.6.)

1
0

b
a
u dv  uv  a   v du
b
b
a
tan 1 x dx
Ex.7.) Find the area of the region bounded by the curve y  xe x and the x-axis on 0, 4 .