1
c
Amy
Austin, January 20, 2016
Section 6.5: Integration by Substitution
The Substitution Rule: If u = g(x) is a differentiable function, then
Z
′
f (g(x))g (x) dx =
Z
f (u) du
Note: Typically, u is chosen so that du is a factor of the integrand.
EXAMPLE 1: Integrals of the form
Z
(f (x))n f ′ (x) dx, if n 6= −1:
If we choose u = f (x), then du = f ′ (x)dx,
hence
Z
n ′
(f (x)) f (x) dx =
a.)
Z
b.)
Z
c.)
Z
Z
un du =
2t2 (t3 − 1)3 dt
sec x tan x(3 + 5 sec x)7 dx
√
x2
dx
1−x
un+1
(f (x))n+1
+C =
+C
n+1
n+1
2
c
Amy
Austin, January 20, 2016
EXAMPLE 2: Integrals of the form
du = f ′ (x)dx, hence
a.)
Z
b.)
Z
Z
Z
ef (x) f ′ (x) dx =
Z
ef (x) f ′ (x) dx. If we choose u = f (x), then
eu du = eu + C = ef (x) + C
e1/x
dx
x2
2
2
xex dx
−1
EXAMPLE 3: Integrals of the form
du = f ′ (x)dx, hence
a.)
Z
1
2
Z
x+1
dx
x2 + 2x
f ′ (x)
dx =
f (x)
Z
Z
f ′ (x)
dx. If we choose u = f (x), then
f (x)
du
= ln |u| + C = ln |f (x)| + C
u
c
Amy
Austin, January 20, 2016
b.)
Z
c.)
Z
x+1
dx
x2 + 1
d.)
Z
tan(x) dx
e4
e3
1
dx
x ln x
3
4
c
Amy
Austin, January 20, 2016
EXAMPLE 4: Integrals of the form
Z
sec(f (x)) tan(f (x))f ′ (x) dx,
Z
Z
′
cos(f (x))f (x) dx,
csc2 (f (x))f ′ (x) dx,
Z
Z
′
sin(f (x))f (x) dx,
Z
sec2 (f (x))f ′ (x) dx,
csc(f (x)) cot(f (x))f ′ (x) dx
Typically, we choose u = f (x), the angle, in these cases.
√
Z
sec2 x
√
a.)
dx
x
b.)
Z
(sin(3α) − sin(3x)) dx
EXAMPLE 5 : How do you choose u? For the following intergals, choose u and
rewrite the integral in terms of u and du.
Z
arctan x
dx
a.)
1 + x2
b.)
Z
√
c.)
Z
x
dx
1 + x4
sin x
dx
1 − cos2 x
5
c
Amy
Austin, January 20, 2016
A few more integrals we added in class:
EXAMPLE 6 : Integrals of the form
NOTE: ALL *LINEAR* u′ s
EXAMPLE 7 :
Z
x3 (1 + x2 )9 dx
Z
ekx dx,
Z
sin(kx) dx,
Z
cos(kx) dx