Chapter 7— Techniques of Integration
Math 1220-004 Class Notes
1. A list of standard integral forms (You should be able to comprehend each
equation from both sides):
R
(1) kdu = ku + C, k constant
 r+1
u
R r
+ C, r 6= −1
(2) u du = r + 1

ln |u| + C, r = −1
R
(3) eu du = eu + C
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
R
R
R
R
R
R
R
R
R
R
au du =
sin udu = − cos u + C
cos udu = sin u + C
sec2 udu = tan u + C
csc2 udu = − cot u + C
sec u tan udu = sec u + C
cscu cot udu = −cscu + C
tan udu = − ln | cos u| + C
cot udu = ln | sin u| + C
√
du
u
= sin−1 ( ) + C
a
a2 − u2
R
a2
R
R
R
au
+C
ln a
du
1
u
= tan−1 ( ) + C
2
+u
a
a
du
1
|u|
1
a
= sec−1 ( ) + C = cos−1 ( ) + C
a
a
a
|u|
u u2 − a2
√
sinh udu = cosh u + C
cosh udu = sinh u + C
1
Section 7.1 Basic Integration Rules
Theorem (Review):
Let g be a differentiable function and suppose that F is an antiderivative
of f . Then if u = g(x),
Z
Z
0
f (g(x))g (x)dx = f (u)du = F (u) + C = F (g(x)) + C
Example 1
Z
3
t
p
t2 − 4dt
2
Example 2
Z
x
cos2 (x2 )
2
dx
Section 7.2 Integration by parts
One of The Most Important Formulas in this Course:
(1) Integration by parts for Indefinite Integrals:
Z
Z
0
u(x)v (x)dx = u(x)v(x) − v(x)u0 (x)dx
(or symbolically)
Z
Z
udv = uv −
vdu
(2) Integration by parts for Definite Integrals:
Z
a
b
¯b Z b
¯
u(x)v (x)dx = [u(x)v(x)]¯¯ −
v(x)u0 (x)dx
0
a
Proof:
3
a
Examples
(1)
(2)
(3)
R
Z
1
x arcsin xdx
0
ln xdx
R5√
1
R
(4)
π
6
π
4
(5)
cos (ln x)dx
2x ln (x3 )dx
(6)
x sec x2 dx
Z
1
4
2
(t + 3)e3t−4 dt
Section 7.3 Some Trigonometric Integrals
We are going to consider five commonly encountered types of Integrals:
(1)
(2)
R
R
sinn xdx and
R
cosn xdx
sinm x cosn xdx
R
R
sin mx cos nxdx, sin mx sin nxdx, cos mx cos nxdx
R
R
(4) tann xdx, cotn xdx
R
R
(5) tanm x secn xdx, cotm x cscn xdx
(3)
R
Type 1:
R
sinn xdx and
R
cosn xdx
Strategy:
(a)For n odd, try (Pythagorean Identity):
cos2 x + sin2 x = 1
(b)For n even, try (Half-Angle Identities):
sin2 x =
1 − cos 2x
2
cos2 x =
1 + cos 2x
2
or
5
Example 1:
Z
cos5 xdx
Example 2:
Z
cos4 xdx
6
Type 2:
R
sinm x cosn xdx
Strategy:
(a)For either m or n odd, try (Pythagorean Identities):
cos2 x + sin2 x = 1
(b)For both m and n even, try (Half-Angle Identities):
sin2 x =
1 − cos 2x
2
cos2 x =
1 + cos 2x
2
or
Example 3
Z
sin3 x cos10 xdx
7
Example 4
Z
sin2 x cos2 xdx
Type 3:
R
sin mx sin nxdx,
R
cos mx cos nxdx,
R
sin mx cos nxdx
Strategy:
Try (Product Identities):
(1)
1
sin mx sin nx = [cos (m − n)x − cos (m + n)x]
2
(2)
(3)
1
cos mx cos nx = [cos (m − n)x + cos (m + n)x]
2
1
sin mx cos nx = [sin (m + n)x + sin (m − n)x]
2
8
Example 5
Z
cos (801x) sin (2014x)dx
Type 4:
R
tann xdx and
R
cotn xdx
Strategy:
Try (Pythagorean Identities):
tan2 x = sec2 x − 1
or
cot2 x + 1 = csc2 x
Example
Z
cot5 3tdt
9
Section 7.4 Rationalizing Substitutions
We are going to study how to deal with three different types of integrands.
(1)
√
n
ax + b
√
(2) (a) a2 − x2
√
(b) a2 + x2
√
(c) x2 − a2
(3) x2 + Bx + C under radicals
Type 1:
√
n
ax + b
Strategy:
Try:
u=
Example 1
√
n
ax + b
Z
2
(5 + x)(1 − x) 3 dx
10
Example 2
Type 2:
√
Z
a2 − x2 ,
√
a2 + x2 ,
√
x 3 x − 4dx
√
x2 − a2
Strategy:
√
π π
a2 − x2 , try: x = a sin t, t ∈ [− , ]
2 2
√
π π
(2) for a2 + x2 , try: x = a tan t, t ∈ (− , )
2 2
√
(3) for x2 − a2 , try: x = a sec t, t ∈ [0, π]
(1) for
Example 3
Z p
a2 − x2 dx
11
Example 4
Z
4
√
2
x2 − 4
dx
x
Type 3: x2 + Bx + C appears under a radical Strategy:
Try: completing the square
Example 5
Z
√
x2
3x
dx
+ 2x + 5
12
Section 7.5 Integration of Rational Functions Using Partial Fractions
Recall that a polynomial function is a function that has the form
y = f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0
Some Examples:
Terminology:
(1) Degree:
(2) Leading coefficient:
(3) Coefficients:
Definition: A rational function is defined as the quotient of two polynomial
functions. So that a rational function has the form:
y = f (x) =
an xn + an−1 xn−1 + · · · + a1 x + a0
bm xm + bm−1 xm−1 + · · · + b1 x + b0
.
Furthermore, if the degree of the numerator is less than that of the denominator, (n < m in this case), then the function is called a proper rational
function, otherwise, an improper rational function.
Some examples of rational functions :
13
A Standard Method to Integrate A Rational Function
To integrate a rational function f (x) =
p(x)
, try the following 6 steps:
q(x)
Step 1: if f (x) is improper, that is, if p(x) is of degree at least that of
p(x), divide p(x) by q(x) by using long division, yielding:
f (x) =a polynomial +
N (x)
D(x)
Step 2: Factor D(x) into a product of linear and irreducible quadratic
factors with real coefficients. By algebra theory, this can always be done,
although we don’t provide proof here.
Step 3: For each factor of the form (ax + b)k , expect the decomposition to
have the terms
A1
A2
Ak
+
+···+
2
(ax + b) (ax + b)
(ax + b)k
Step 4: For each factor of the form (ax2 +bx+c)m , expect the decomposition
to have the terms
B1 x + C1
B2 x + C2
Bm x + Cm
+
+···+
ax2 + bx + c (ax2 + bx + c)2
(ax2 + bx + c)m
N (x)
equal to the sum of all the terms found in Step 3 and
D(x)
Step 4. The number of constants to be determined should equal the degree
of the denominator, D(x).
Step 5: Set
Step 6: Multiply both sides of the equation found in Step 5 by D(x) and
solve for the unknown constants. This can be done by either of two methods:
(1) Equate coefficients of like-degree terms or (2) assign convenient values
to the variable x.
14
(1)
Z
x6 + 4x3 + 4
dx
x3 − 4x2
15
(2)
Z
x3
dx
x2 + x − 2
16
(3)
Z
1
(x −
1)2 (x
17
+ 4)2 dx
(4)
Z
2x2 − 3x − 36
dx
(2x − 1)(x2 + 9)
18