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Calculus II
Strategies for Integration (Review section)
Although the order in which you try various methods is not fixed, I find the following helpful.
There is no substitution for practice when it comes to quickly identifying the proper method.
1. Can the integrand first be simplified using basic algebra, logarithm properties or trigonometric identities?
2. Will “regular” Calculus I u-substitution work?
3. Does the integrand contain multiplication that will make Integration-by-Parts the best method?
4. Does the integrand contain a single function whose derivative is known or a product or quotient where you know the derivative of
one function and the other is a polynomial that’d be easy to integrate? If yes, then try Integration-by-Parts.
5. Does the integrand contain powers of trigonometric functions or products of powers? Then use the method that applies to the
particular trigonometric function and even/odd power.
6. Does the integrand contain a  x or
triangle to find the substitutions.
2
2
x  a or
2
2
x  a ? Then use trigonometric substitution, i.e. draw the appropriate
2
2
7. Does the integrand contain a quotient of two polynomials where the degree on the top is greater than or equal to the degree on the
bottom? If yes, then use long division first.
8. Does the integrand contain a quotient of two polynomials where the degree on the top is smaller than the degree on the bottom? If
yes, then use partial fractions.
9. Does the integrand contain division, but not of polynomials? Can you use u-substitution to convert the integrand into a quotient of
polynomials?
10. If none of the above seem to work, how about a variation of u-substitution where you solve for u, like in #9?
For the following problems, without doing any calculus, decide which of the above 10 methods you think is
most appropriate to start with.
1.
x
dx
x 5
6.
x2
dx
e3 x
2. x (1  x )dx
2
7.
e
3. ln xdx
4. ln xdx
5. ( x  4) 2 ln xdx
x
x
dx; There are two similar ways to do this. Can you find both variations?
tan 1 x
dx; Eventually, this will require two methods to complete.
x2
x 1
x2 1
dx
9.
dx
10. 2 dx
11. 2
x 1
x 1
2 x  5x  2
8.
13.
x
dx
14.
x2 1
dx
x
15. cos 2  sin  d
12.
x2  2x  1
dx
( x  1) 2 ( x 2  1)
x 1
tan 
16.
d ; Eventually, you will have a choice of 3 ways to do this one. Can you find all three ways?
sec 2 
cos x
sec 2 
2
17. x sin xdx
18. sin x dx
19.
d
20. tan 7 x sec3 xdx
e
1  tan 
3
sin x
21.
dx; Eventually, this will require two methods to complete.
cos x
2