Download How to Attack an Integral We have six basic techniques of

How to Attack an Integral
We have six basic techniques of integration. Remember that it is possible to evaluate some integrals
using more than one method.
1. Simple Substitution - Look for an obvious substitution. It is important to be able to
recognize functions with their derivatives:
e.g. sin x with cos x dx
tan x with sec2 x dx
ln x with
1
x
dx
2. Trigonometric Identities - If the integrand contains powers of trig functions, then try
splitting off a cos x, sin x, sec x tan x, sec2 x, csc x cot x, or csc2 x. Then look for a substitution.
If this doesn’t work, then you may want to try using one of the following identities.
cos2 x + sin2 x = 1
tan2 x + 1 = sec2 x
cot2 x + 1 = csc2 x
1
[1 + cos 2x]
2
1
sin2 x = [1 − cos 2x]
2
cos2 x =
3. Trigonometric Substitution - If the integrand contains any of the expressions a2 − x2 ,
a2 + x2 , or x2 − a2 , then try the appropriate trig substitution (x = a sin θ, x = a tan θ, or
x = a sec θ, respectively).
4. Completing the Square - If the integrand contains the square root of a quadratic expression
OR if the integrand contains a rational function whose denominator is an irreducible quadratic
expression, then you may be able to complete the square and use trig substitution.
5. Partial Fractions - If the integrand is a rational function, then factor the denominator into
linear and irreducible quadratic factors and find the partial fraction decomposition. Remember
to divide first if the degree of the numerator is greater than or equal to the degree of the
denominator.
6. Integration by Parts - This method works best when the integrand contains a product of
functions. Remember that you may have to apply i.b.p. more than once.
Advice from George F. Simmons: “Be observant, thoughtful, flexible and persistent—all of which
are of course easier said than done. If a method doesn’t work, be ready to try another. Sometimes
several methods work. Keep your options open and do things the easy way— if any. And remember
that doing a problem more than one way is a good learning experience.”
2
Use this strategy to evaluate the following integrals. You will not be required to turn these in but
you may ask questions about them in office hours.
Z
Z
4
1.
3
sin x cos x dx
Z
Z
dx
q
4.
2.
3
5.
dx
√
6x − x2
p
ln 1 + x2 dx
Z
3.
Z
6.
(a2 + x2 )
Z
7.
Z
10.
Z
13.
3x − 7 dx
(x − 1) (x − 2) (x − 3)
Z
8.
11.
tan−1 10x dx
14.
Z
Z
22.
tan 3x sec 3x dx
17.
dx
x2 + 4x + 8
20.
x5 dx
√
1 + x2
23.
Z
31.
Z
34.
9.
√
x dx
√
x
12.
2
Z
15.
x3 + x2
dx
x2 + x − 2
dx
4
x + 4x2 + 3
dx
x (2 + ln x)
Z
x
e cos 2x dx
18.
x2 sin x dx
21.
cos x dx
sin3 x − sin x
24.
Z
26.
29.
Z
32.
Z
35.
x + 1 dx
x2 (x − 1)
³
´2/3
x2/3 x5/3 + 1
dx
4x + 1
dx
x3 + 4x
Z
√
x ln x + 2 dx
Z
x2 ex dx
Z
dx
3/2
Z
tan x sec x dx
28.
dx
(x2 − 1)
Z
2
Z
cos
sin−1 x dx
√
1 − x2
Z
x
Z
Z
25.
√
Z
3
19.
e
Z
tan x dx
cos2 x
Z
16.
ln
cos x dx
√
1 + sin x
dx
x1/5
x3 dx
√
16 − x2
cot x dx
ln (e sin x)
sin4 x cos2 x dx
Z
x tan2 x dx
Z
27.
x dx
− 16
x4
Z
30.
sin x cos 2x dx
Z
sin3 2x cos3 2x dx
33.
Z
36.
sec2 x dx
sec2 x − 3 tan x − 1