7.1 - 7.5 Review - Integration Techniques
My reviews and review sheets are not meant to be your only form of studying. It is vital to your success
on the exams that you carefully go through and understand ALL the homework problems, worksheets
and lecture material. Hopefully this review sheet will remind you of some of the key ideas of these
sections.
1. Integration by Parts
•Integration by Parts Formula: Rudv =uv −Rvdu and Rb
audv =uv|b
a−Rb
avdu
•Understand how to perform integration by parts and how to choose your u. Remember u
should be ‘easy’ to differentiate and dv should be ‘easy’ to antidifferentiate. To help you
decide we discussed LIPET (Logs, Inverse trig, Powers of x, Exponentials, Trig); let uthe
part of the integrand that occurs earlier in LIPET.
•There is no one rule that tells you when to use which methods. But here are some instances
when integration by parts tends to work:
–Products - specifically powers of xtimes sin(x), cos(x), or ex. It also works with ex
times sin(x) or cos(x) (but you have to pay attention to when the process repeats).
–Logs - integrals involving logs (take u= ln(x))
–Inverse Trig - integrals involving inverse trigonometric functions (take u= ‘inv. trig’)
2. Trigonometric Integrals
•There is one general theme: Rewrite the integral using identities and use u-substitution (if
needed).
•The identities that are most useful are:
cos2(x)+sin
2(x) = 1 , which we may use as cos2(x)=1−sin2(x) or sin2(x)=1−cos2(x)
Dividing these by cos2(x) gives more useful identities:
1 + tan2(x) = sec2(x) , which we may use as 1 = sec2(x)−tan2(x) or tan2(x) = sec2(x)−1
We will also need the half angle identities:
sin2=1
2−1
2cos(2x) , cos2(x)= 1
2+1
2cos(2x) , and sin(x) cos(x)= 1
2sin(2x)
•Page 484 and 486 discuss strategies for when to use which identities and why. For me, it
is most useful to attempt to set up a u-substitution, this gives three possibilities for sin(x)
and cos(x) and three possibilities for sec(x) andtan(x):
•Integrals with sin(x) and cos(x)
(a) Using u= sin(x) and du = cos(x)dx, look for an odd power of cos(x) and pull out one
factor. Then use the identites to turn the rest of the problem into sin(x)’s.
(b) Using u= cos(x) and du =−sin(x)dx, look for an odd power of sin(x) and pull out
one factor. Then use the identites to turn the rest of the problem into cos(x)’s.
(c) If both powers are even, the above two methods don’t work. Use the half angle identities
to simplify the problem.
•Integrals with sec(x) and tan(x)
(a) Using u= tan(x) and du = sec2(x)dx, look for an even power of sec(x) and pull out
two factors. Then use the identites to turn the rest of the problem into tan(x)’s.
(b) Using u= sec(x) and du = sec(x) tan(x)dx, look for an odd power of tan(x) and pull
out one factor of tan(x) and one factor of sec(x). Then use the identites to turn the
rest of the problem into sec(x)’s.
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