1. integration is more challenging than differentiation.
Sec 7.5 : STRATEGY FOR INTEGRATION
integration is more challenging than differentiation.
No hard and fast rules can be given as to which method applies in a given situation, but we give some advice on strategy that you may find useful.
how to attack a given integral, you might try the following four-step strategy.

2. 1 2 3 4 4-step strategy Sec 7.5 : STRATEGY FOR INTEGRATION
Simplify the Integrand if Possible 2
Look for an Obvious Substitution
function and its derivative 3
Classify the Integrand According to Its Form
Trig fns, rational fns, by parts, radicals, 4
Try Again
1)Try subsitution 2)Try parts 3)Manipulate integrand
4)Relate to previous Problems 5)Use several methods

3. 1 4-step strategy Sec 7.5 : STRATEGY FOR INTEGRATION
Simplify the Integrand if Possible

4. 2 4-step strategy Sec 7.5 : STRATEGY FOR INTEGRATION
Look for an Obvious Substitution
function and its derivative

5. 3 4 4-step strategy Sec 7.5 : STRATEGY FOR INTEGRATION
Classify the integrand according to Its form
Trig fns, rational fns, by parts, radicals, rational in sine and consine
7.2
7.4
7.1
7.3
7.4 4
Try Again
1)Try subsitution 2)Try parts 3)Manipulate integrand
4)Relate to previous Problems 5)Use several methods

6. 3 1 2 3 4 6 7 5 8 9 Sec 7.5 : STRATEGY FOR INTEGRATION
Classify the integrand according to Its form 1 2
Integrand contains:
Integrand contains:
ln and its derivative
f and its derivative
by parts
by parts
We know how to integrate all the way 3 4
Integrand =
Integrand radicals:
by parts (many times)
7.3
PartFrac 6
Integrand = rational
Radicals with different indexes
f & f’ 7
Back to original
2-times by part  original 5
Integrand contains: only trig
7.2 8 9
Rational sine & cosine:
Combination:

7. Sec 7.5 : STRATEGY FOR INTEGRATION
132
141
121
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
Rat in sin,cos
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
several
original
Back

8. Sec 7.5 : STRATEGY FOR INTEGRATION
132
141
121
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
Rat in sin,cos
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
several
original
Back

9. Sec 7.5 : STRATEGY FOR INTEGRATION
elementary functions.
polynomials,
rational functions
power functions
Exponential functions
logarithmic functions
trigonometric
inverse trigonometric
hyperbolic
inverse hyperbolic
all functions that obtained from above by 5-operations
FACT:
If g(x) elementary
g’(x) elementary
NO:
If f(x) elementary
need not be an elementary

10. Sec 7.5 : STRATEGY FOR INTEGRATION
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
YES or NO if
Continuous.
Anti-derivative
exist?
Will our strategy for integration enable us to find the integral of every continuous function?
YES or NO

11. Sec 7.5 : STRATEGY FOR INTEGRATION
elementary functions.
polynomials,
rational functions
power functions
Exponential functions
logarithmic functions
trigonometric
inverse trigonometric
hyperbolic
inverse hyperbolic
all functions that obtained from above by 5-operations
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
YES
Will our strategy for integration enable us to find the integral of every continuous function? NO

12. Sec 7.5 : STRATEGY FOR INTEGRATION
FACT:
If g(x) elementary
g’(x) elementary
NO:
If f(x) elementary
need not be an elementary
has an antiderivative
is not an elementary.
This means that no matter how hard we try, we will never succeed in evaluating in terms of the functions we know.
In fact, the majority of elementary functions don’t have elementary antiderivatives.

13. Sec 7.5 : STRATEGY FOR INTEGRATION
Example
Let
Evaluate the following integrals in terms of F

14. Sec 7.5 : STRATEGY FOR INTEGRATION

16. Sec 7.5 : STRATEGY FOR INTEGRATION
102
122
111
112
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

17. Sec 7.5 : STRATEGY FOR INTEGRATION
102
122
111
112
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

18. Sec 7.5 : STRATEGY FOR INTEGRATION
142
141
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

19. Sec 7.5 : STRATEGY FOR INTEGRATION
142
141
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

20. Sec 7.5 : STRATEGY FOR INTEGRATION
132
131
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

21. Sec 7.5 : STRATEGY FOR INTEGRATION
132
131
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

22. Sec 7.5 : STRATEGY FOR INTEGRATION
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

23. Sec 7.5 : STRATEGY FOR INTEGRATION
Partial fraction
Simplify integrand
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

24. Sec 7.5 : STRATEGY FOR INTEGRATION
Partial fraction
Simplify integrand
others
Radicals(Trig subs)
f & f’
conjugate
radical with diff index
Trig fns
by parts
Power of
Others(identity)
Obvious subs
Rational in sin & cos
several
original
Back

27. Example
Example

28. Example