Download AP Calculus Section 4 - LCMR School District

AP Calculus Section 6.3
Integration by Substitution
Homework:
Page 371 #9 – 45 odd and #63, 65
Objective: SWBAT solve integration problems using u-substitution.
The method of substitution can be analyzed by looking at the chain rule.
We know by the chain rule that
d
 F ( g ( x))  F '  g ( x)  g '( x)
dx
Therefore
 F '  g ( x)  g '( x)dx  F ( g ( x))  C
If this is true, why? This will lead us into integral substitution.
To integrate using u-substitution do the following:
a. Make a choice for u, say u  g ( x )
du
 g '( x) and solve for dx
b. Compute
dx
c. Make the substitution :
u  g ( x)
du
g '( x)
At this point, the entire integral should be in terms of u. If it is not you need to
rethink u, or make an additional substitution.
dx 
d. Evaluate the integral and replace u with g ( x )
x
2
 1
50
 sin  x  9 dx
 2 x  dx
u=
u=
 cos  5x dx
 sin
2
x cos xdx
u=

e
x
x
u=

dx
u=
dx
1
3
x  8
5
u=
t
4
x
3  5t 5 dt
2
x  1dx
u=
u=
1

2
  x  sec  x dx
x2  2
 x  1 dx
u=
u=
5
cos 3x
 3x  2dx
 2  sin 3x dx
u=
u=
 sin
e2 x
 e2 x  4dx
3
x cos 2 xdx
u=
u=
 cos
3
 sin
xdx
5
x cos 2 xdx
u=

u=
x
1
sin xdx
x
x  5dx
u=
u=
Inverse Trig Integrals – We will focus on only these two inverse integrals.
a

2
du
1
u
 tan 1  c
2
u
a
a
du
a u
2
2
 sin 1
u
c
a
1. Find the integral
dt
 9  4t
2
2. Find the integral

dt
1  4t 2
Extra Practice Section 6.3 u-Substitution
1.
 3x
2.

3.
 cos
4.
 csc 2t cot 2tdt
5.
x
6.
 cos
7.

8.
 tan 3x sec
9.

x 2 dx
x 1
10.

x 2 dx
11.
 9  4t
12.

13.
x
1  2 x 2 dx
3 xdx
3  7 x2
3
dx
2
2x
5 x 4  18dx
3
sin x
dx
3
x
1
sin xdx
x
1  x3
dt
2
dx
1  4x2
4
x
dx
9
2
3 xdx