Math 1206 Calculus – Chapter 7 Techniques of Integration
Sec. 7.1: Integration by Parts
I.
Introduction
If the techniques/ formulas we introduced earlier do not work then there is another technique
that you can use: Integration by Parts. It is a way of simplifying integrals of the
form:
∫ f ( x ) g( x ) dx
in which f can be differentiated repeatedly and g can be integrated
repeatedly without difficulty.
II.
The Formula
A. Derivation of the Formula
The formula for integration by parts comes from the Product Rule.
d
dv
du
(uv ) = u
+ v
dx
dx
dx
d (uv ) = u dv + v du ; (differential form)
u dv = d (uv ) − v du ; (solve for u dv )
∫ u dv
= uv −
∫ v du
; (integrating)
B. Note
The integration by parts expresses one integral,
∫ u dv , in terms of another, ∫ v du .
The
idea is that with a proper choice of u and v, the second integral is easier to integrate than
the original. You may have to use this technique several times before you reach an
integral that you can integrate easily.
C. The Integration-by-Parts Formula
1. Indefinite Integrals
If u and v are functions of x and have continuous derivatives, then
∫ u dv
= uv - ∫ v du
2. Definite Integrals
v2
∫ u dv
u2
= uv
v1
u1
u2
- ∫ v du
u1
D. How to pick u and dv
When deciding on your choice for u and dv, use the acronym: ILATE with the higher one
having the priority for u.
I
L
A
T
E
-
inverse trig fns
logarithmic fns
algebraic fns
trigonometric fns
exponential fns
I I I . Examples
1.
∫ 3xe
2.
∫ ln x dx
2x
dx
e
3.
∫x
1
2
ln x dx
3
4.
∫ tan
−1
x dx
1
5.
∫ye
2 3y
dy
6.
∫t
7.
∫e
4
x
sin(2t ) dt
cos(4x ) dx
8. Use the reduction formula
∫ sec
4
y dy
∫ sec
n
1
u du = n−1
tan u sec n−2 u + n−2
sec n−2 u du to evaluate
n−1 ∫
9.
∫t
3
( )
cos t 2 dt
10. Find the area of the region enclosed by the curve y=xcosx, the x-axis from
x=
2
to x =
3
.
2
11. Find the volume of the solid generated by revolving the region in the 1st quadrant bounded
by the axes, the curve y=ex and the line x=ln2 about the line x=ln2.
IV.
A Summary of Common Integrals Using Integration by Parts:
∫ x e dx , ∫ x sin (ax ) dx , ∫ x cos(ax ) dx ,
∫ x ln x dx , ∫ x sin ( ax ) dx , ∫ x tan ( ax ) dx ,
∫ e sin( bx ) dx , ∫ e cos(bx ) dx
n ax
n
ax
n
n
n
−1
ax
n
−1