Integral formulas and identities you need to know.
R
n+1
1. xn dx = xn+1 + C (n 6= −1)
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
1
x
dx = ln|x| + C
ex dx = ex + C
ax dx =
ax
lna
+C
sin x dx = − cos x + C
cos x dx = sin x + C
sec2 x dx = tan x + C
csc2 x dx = − cot x + C
sec x tan x dx = sec x + C
csc x cot x dx = − csc x + C
sec x dx = ln| sec x + tan x| + C
csc x dx = ln| csc x − cot x| + C
tan x dx = ln| sec x| + C
cot x dx = ln| sin x| + C
dx
x2 +a2
dx =
√ dx
a2 −x2
1
a
tan−1 ( xa ) + C
dx = sin−1 ( xa ) + C
17. cos2 (x) = 12 (1 + cos 2x)
18. sin2 (x) = 12 (1 − cos 2x)
19. sin2 x + cos2 x = 1
20. tan2 x + 1 = sec2 x
21. 1 + cot2 x = csc2 x
22. sin 2θ = 2 sin θ cos θ
Outline of SOME of the concepts you need to know for Test 3
For every section you should consult homework problems and notes for practice problems. The problems listed below are not enough to study for the exam, they are listed more to illustrate a concept
too difficult to explain in words.
Section 5.5-5.6
• U-Substitution, know it, love it, look for it after you have simplified an integral.
• BE VERY CAREFUL ABOUT THE LIMITS OF INTEGRATION!!! Either change them to
match your u-sub or substitute your u back in. I recommend changing them, then you don’t need
to substitute your u back in.
• All of the following (although it is not an
can do it in your
R inclusive
R list) require
R a u-sub, Rif you
1
head that’s fine otherwise write it out! eax dx, cos axdx, sin axdx, ax+b
dx. Where a and
b are constants.
Section 7.1 Integration by parts
R
R
• Know udv = uv − vdu
• Have an intuition as to what to choose for your “u” remember ILATE.
• Remember the “classic cases” as outlined on the integration by parts handout posted on my home
page.
Section 7.2 Trigonometric Integrals
• Know the different cases, what you should “save”, what identity you should substitute and what
u-sub you should make after you’ve used an identity. Also
your half-angle formulas (17 and
R know
n
18 on the first page) which are useful when you have
sin
(x)
cosmR(x) dx, where n and m are
R
2
both even. You should definitely know how to find sin (x) dx and cos2 (x) dx.
Section 7.3 Trigonometric substitution
• Know what substitutions to make and what identities to use to simplify. You should be able to
immediately recognize integrals that require a trig. sub.
• Make sure you don’t forget to properly change the dx into dθ. For example, if x = tan(θ), then
dx = sec2 (θ)dθ.
• Make sure you can write the answer in terms of the original variable when you finish the integration
step. You should be able to use your triangle to write any trig. function in terms of the original
variable.
• Your answer may contain a sin(2θ). If so, use formula (22) on page 1 to write sin(2θ) =
2 sin(θ) cos(θ). Then you should use your triangle to solve for these quantities in terms of the
original variable. Problems 7 and 25 from section 7.3 homework illustrate this.
Section 7.4 Partial Fractions
• Know the rules for decomposition.
• Be able to long divide. You need to long divide if the highest power in the numerator is greater
than or equal to the highest power in the denominator.
• Factor your denominator as much as it can be. You can always factor into products of repeated
or non-repeated linear factors and repeated or non-repeated irreducible quadratics. Problem 19
from section 7.4 homework is a good problem illustrating the decomposition. If you do not factor
completely you will not be writing the correct partial fraction decomposition.
• Once you Rhave done your partial fraction decomposition, you will have to integrate terms that
1
look like (ax+b)
You also can have terms that look like
n dx which you do by substitution.
R Ax
R
R Ax
R Ax+B
the integral up into x2 +a2 dx and x2B
x2 +a2 dx. Here you split
+a2 dx.
x2 +a2 dx is done
R B
using a substitution and x2 +a2 dx is done using formula (15) on page 1.
• You will not be expected to complete the square on this test.
Section 7.5 Tables of integration
• Know how to manipulate an integral so that it matches a formula in the table and be able to
evaluate.
Section 7.6 Numerical integration
• Know the trapezoidal rule and Simpson’s rule on pages 479 and 481 respectively. Know how to
use the error bounds associated with the rules on page 481.
• Know how to use the error bounds to figure out how large an “n” you need to chose to get within
a certain error range.
Strategy for Integration
1. SIMPLIFY!!!
2. Look for a u-sub
3. Complete the square
4. Use Trig Identities or a trig substitution
5. Reduce improper fractions (long divide)
6. Use partial fractions
7. Integrate by parts
NOTE: Once you do one of these things you, if you still can’t integrate start at the top of the list again and
see what you can do.
Generally speaking, I will not tell you what method to use to find a given integral. You have to figure out
what method is applicable yourself. You need to be comfortable with all of the techniques and see obvious
cases in which they apply. You do not want to waste too much time remembering formulas or figuring out
which method to use in an obvious case.
Sometimes it is not obvious how to proceed. Trial and error is ok as long as you do correct math. A good
example is problem (1) from the extra credit worksheet assigned recently. You have to do something
√ to
x or
get rid
of
the
square
root,
so
try
a
substitution.
As
long
as
you
do
the
substitution
properly,
u
=
√
u = x + 10 both lead you to a new integral you can do using a different method.