This document discusses integration by substitution. It provides examples of using substitution to evaluate various integrals involving trigonometric, exponential, and logarithmic functions. Guidelines are provided for choosing the substitution variable u. Several worked examples demonstrate how to use substitution to rewrite integrals in terms of u and then evaluate the integral. Exercises at the end provide additional practice problems for students to evaluate using integration by substitution.

1. Calculus-II
Math-102
Dr. Farhana Shaheen
5.3 Integration by substitution

2. Always Remember:
Integration is by
Observation
 And…
 The Best way is by
Substitution

3. Integrals of Transcendental
Functions
 Trigonometric functions
 The inverse trigonometric functions
 Exponential functions
 Logarithms.

10. C
1n
)x(f
dx)x(f)x(f
C)x(fln
)x(f
dx)x(f
:RULESNINTEGRATIOIMPORTANT
1n
n









11. INTEGRATION BY SUBSTITUTION
Note:
Integration by substitution can be used for a variety of
integrals: some compound functions, some products and
some quotients.
Sometimes we have a choice of method.

12. Choosing u

13. Guidelines for u-Substitution (p. 334)

14. (Page: 337)
Equation (5)
Equation (6)
Equation (7)

16. The chain rule allows us to
differentiate a wide variety of
functions, but we are able to find
antiderivatives for only a limited
range of functions? We can
sometimes use substitution or
change of variable to rewrite
functions in a form that we can
integrate.


17. Example 1:
 
5
2x dx Let 2u x 
du dx
The variable of integration
must match the variable in
the expression.
Don’t forget to substitute the value
for u back into the problem!

 duu5
cu  6
6
1
c
x



6
)6( 6

18. Example 2:
2
1 2x x dx 
One of the clues that we look for is
if we can find a function and its
derivative in the integral.
The derivative of is .
2
1 x 2x dx
1
2
u du
3
2
2
3
u C
 
3
2 2
2
1
3
x C 
2
Let 1u x 
2du x dx
Note that this only worked because
of the 2x in the original.
Many integrals can not be done by
substitution.


19. Example 3:
4 1x dx Let 4 1u x 
4du dx
1
4
du dx
Solve for dx.
1
2
1
4
u du
3
2
2 1
3 4
u C 
3
2
1
6
u C
 
3
2
1
4 1
6
x C 


20. Example 4:
 cos 7 5x dx
7du dx
1
7
du dx
1
cos
7
u du
1
sin
7
u C
 
1
sin 7 5
7
x C 
Let 7 5u x 


21. Example 5:
 2 3
sinx x dx
3
Let u x
2
3du x dx
21
3
du x dx
We solve for
because we can find it
in the integral.
2
x dx
1
sin
3
u du
1
cos
3
u C 
31
cos
3
x C 


22. Example 6:
4
sin cosx x dx
Let sinu x
cosdu x dx
 
4
sin cosx x dx
4
u du
51
5
u C
51
sin
5
x C


23. Example: 7

24. Example: 8

25. Example: 10 (page 336)
Evaluate:
dx
e1
e
x2
x
 

26. Example: 11 (page 336)
Evaluate:
dx1xx2


27. Exercises
Use substitution to integrate the following. (Where
possible, you could also use a 2nd method.)
  dxx 8
)1(1. 

dxe x3
2.
  dxxx 42
)1(3.

28. Question for you:

29. Ex: 5.3 Questions: 1 – 54 (page 338)
Questions: 1 – 12 (substitution is given)
Questions: 15 – 54 (using substitution)
Questions: 61 – 62 (using substitution and
formulas)