13.2 Integration by
Substitution
• Let w be the inside function
• Write
dw
 w' ( x)
dx
• Do cross-multiply
• Back to the given integral and do the
substitution
• Take the integral
• Express the final answer in term of x
(
2
x

3
)
(
6
x
)
dx

?

3
20
2
 (2 x
 w
3
dw
 6x 2
dx
dw  6 x dx
2
 3) (6 x )dx
20
20
dw
2
5
e
dx

?

5x
5
e
dx

5x
dw
5
dx
  e dw
dw  5dx
 e C
w
w
 e C
5x
1
 4  x 2 2 xdx  ?
dw
 2x
dx
dw  2 xdx
1
 4  x 2 2 xdx
1
  dw
w
 ln w  C
 ln 4  x  C
2
(
x

3
x

7
)
(
2
x

3
)
dx

?

2
4
 ( x  3 x  7)
  w dw
2
dw
 2x  3
dx
dw  (2 x  3)dx
4
( 2 x  3)dx
4
5
w

C
5
( x  3 x  7)

C
5
2
5
(
2
x

3
)
(
6
x
)
dx

?

3
4
2
(
2
x

3
)
(
6
x
)
dx

3
4
2
  w dw
4
dw
2
 6x
dx
dw  6 x dx
2
5
w

C
5
(2 x  3)

C
5
3
5
e

dw
 3
dx
dw  3dx
dw
 dx
3
1
dw  dx
3
3 x
dx  ?
3 x
e
 dx
1
  e  dw   1 e w dw
3
3
w
1 w

e C
3
 1 3 x

e C
3
x
dx
2
 x 9
dw
 2x
dx
dw  2 xdx
dw
 xdx
2
1
dw  xdx
2
x
 x 2  9dx
1 1
   dw
w 2
1 1
   dw
2 w
1
 ln w  C
2
1
 ln x 2  9  C
2
 0.001e
0.01x
dx  ?
.01x
.
001
e
dx

dw
 .01
dx
dw  .01dx
dw
 dx
.01
100dw  dx
  .001e w (100dw)
  0.1e dw
w
 0.1e  C
w
 0.1e
.01x
C
• Yah! We are done.
• What have you studied in MAT 116?
• What is Calculus?
• What is “calculus”?
– Calculus is the study of how things
change.
• What do you learn in a calculus class?
– Functions, limits, derivatives, integrals, …
• Derivatives can used to study applications that
involve the rate of change such as population
growth, radioactive decay, optimization
problems, …
• Integrals can be used to calculate the area,
equation of a curve, business sales, …
• How do algebra and calculus differ?
Ex:
Solving equations
to find a particular
value of a variable
Ex:
How a change in a variable affects
another variable?