1. Sec 5.5
SYMMETRY
THE SUBSTITUTION RULE

2. SYMMETRY Sec 5.5: THE SUBSTITUTION RULE
Suppose f is continuous on [-a, a] and even
even
Suppose f is continuous on [-a, a] and odd
Odd

3. Sec 5.5: THE SUBSTITUTION RULE
Term-102

4. Term-091 Sec 5.5: THE SUBSTITUTION RULE (even) + (even) = (even)
(odd) +(odd ) = (odd )
(even) X (even) = (even)
(odd ) X (odd ) = (even)
(even) X (odd ) = (odd )

5. Sec 5.5: THE SUBSTITUTION RULE
Term-103

6. SYMMETRY Sec 5.5: THE SUBSTITUTION RULE
Suppose f is continuous on [-a, a] and even
Suppose f is continuous on [-a, a] and odd
Example

7. Sec 5.5: THE SUBSTITUTION RULE

8. Sec 5.5: THE SUBSTITUTION RULE

9. Sec 5.5
THE SUBSTITUTION RULE

10. Sec 5.5: THE SUBSTITUTION RULE
Find
Table Indefinite Integrals
Find
Find
The Substitution Rule

11. Definite Integral Sec 5.5: THE SUBSTITUTION RULE Example
evaluate the expression in u
Example
return to the variable x

12. Sec 5.5: THE SUBSTITUTION RULE
Table Indefinite Integrals
Find
The Substitution Rule

13. Sec 5.5: THE SUBSTITUTION RULE

14. Sec 5.5: THE SUBSTITUTION RULE
Main Idea
In general, this method works whenever we have an integral that written as a product of function and its derivative.
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Simple Integral
The idea behind the Substitution Rule is to replace a relatively complicated integral by a simpler integral. This is accomplished by changing from the original variable to a new variable that is a function of .
Main Challenge
The main challenge in using the Substitution Rule is to think of an appropriate substitution. You should try to choose to be some function in the integrand whose differential also occurs (except for a constant factor).
Try Another
Finding the right substitution is a bit of an art. It’s not unusual to guess wrong; if your first guess doesn’t work, try another substitution.

15. 132 Sec 5.5: THE SUBSTITUTION RULE
Poly. with high power & its derivative
u = poly
132

16. 082 Sec 5.5: THE SUBSTITUTION RULE
Poly. In denominator and its derivative in the nominator
u = poly
082

17. 092 141 Sec 5.5: THE SUBSTITUTION RULE f (sin x) & cos x dx u = sin
f (cos x) & sin x dx
u = sin
092
141

18. Sec 5.5: THE SUBSTITUTION RULE
ln(x)& 1/x dx
u = ln(x)
T-102

19. Sec 5.5: THE SUBSTITUTION RULE
tan(x)& sec^2(x) dx
u = tan(x)
082

20. Sec 5.5: THE SUBSTITUTION RULE
e^x & e^x dx
a^x & a^x dx
092
121

21. Sec 5.5: THE SUBSTITUTION RULE
Sqrt(---)

22. Sec 5.5: THE SUBSTITUTION RULE
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23. Sec 5.5: THE SUBSTITUTION RULE

24. Sec 5.5: THE SUBSTITUTION RULE

26. Sec 5.5: THE SUBSTITUTION RULE

27. Sec 5.5: THE SUBSTITUTION RULE

28. Sec 5.5: THE SUBSTITUTION RULE
082

29. Sec 5.5: THE SUBSTITUTION RULE
082

30. Sec 5.5: THE SUBSTITUTION RULE
082
T-102

31. Sec 5.5: THE SUBSTITUTION RULE
Find
Example
Example