Chapter 5. Section 5
Page 1 of 3
C. Bellomo, revised 14-Apr-09
Section 5.5 – Substitution
A Look Back:
• Recall the chain rule, where for example 22
(2 )
xx
deex
dx =.
• Clearly, since we know integrals undo derivatives, we would have 22
(2 )
xx
exdxec=+
∫
If we let 2
ux=, we would get 2 , or 2
du
x
du x dx
dx ==.
Substituting u and du into the integral, we would have 2
uu x
edu e c e c
=
+= +
∫
• This ‘method’ is the chain rule used backwards.
• You CANNOT evaluate an integral of a product (like above) without using substitution.
• Formally, the definition of the substitution rule is ( ()) () ()
f
g x g x dx f u du
′=
∫
∫ where u = g(x)
Substitutions with Constants:
• If g(x) = cx, then setting u = cx yields , or
du du
cdx
dx c
==
• Example. Evaluate 2x
edx
∫
Let u = 2x. Then dx = du/2.
2
2
2
1
2
1
2
xu
u
x
du
edx e
ec
ec
=
=+
=+
∫∫
Other Substitutions Require More Pieces:
• Example. Evaluate 22
(1)
xdx
x+
∫
Let 21ux=+. Then 2, 2
du
du xdx xdx==
22 2
2
1
2
11
(1) 2
1
2
1
21
1
2( 1)
du
xdx
xu
udu
uc
c
x
−
−
=
+
=
=+
−
−
=+
+
∫∫
∫
