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Math 222 Antiderivatives of Functions and Some Formula
Table of Indefinite Integrals
R n
R
x+1
x dx = xn+1 (n 6= −1)
1
x dx
= ln |x|
ax
ln a
R
ex dx = ex
R
ax dx =
R
sin xdx = − cos x
R
cos xdx = sin x
R
sec2 xdx = tan x
R
csc2 xdx = − cot x
R
sec x tan xdx = sec x
R
csc x cot xdx = − csc x
R
sec xdx = ln | sec x + tan x|
R
csc xdx = ln | csc x − cot x|
R
tan xdx = ln | sec x|
R
cot xdx = ln | sin x|
R
sinh xdx = cosh x
R
cosh xdx = sinh x
R
1
dx
x2 +a2
tan−1 ( xa )
R
√ 1
dx
a2 −x2
= sin−1 ( xa )
ln x−a
x+a R
√ 1
dx
x2 ±a2
√
= ln x2 ± a2 R
1
x2 −a2
=
dx =
1
a
1
2a
I Techniques of Integration
(A) Trigonometric functions
R
(1) sinm x cosn xdx
- If n is odd, save one cosine and use cos2 x = 1 − sin2 x, then let u = sin x.
- If m is odd, save one sine and use sin2 x = 1 − cos2 x, then let u = cos x.
- If m, n are even, use sin2 x = 21 (1 − cos 2x), cos2 x = 21 (1 + cos 2x) or sin x cos x =
1
2
sin 2x.
R
(2) tanm x secn xdx
- If n is even, save one sec2 and use sec2 x = 1 + tan2 x, then let u = tan x.
- If m is odd, save one sec x tan x and use tan2 x = sec2 x − 1, then let u = sec x.
(B) Rational functions: f (x) =
P (x)
Q(x)
- If f is improper, that is, degP (x) ≥deg Q(x), use long division.
- If f is proper, that is, degP (x) <deg Q(x), then
(1)The denominator Q(x) is a product of distinct linear factors:
(2) Q(x) is a product of linear factors, some are repeated:
x2 +2x−1
x(2x−1)(x+2)
x3 −x+1
x(x−1)3
(3)Q(x) contains irreducible quadratic factors, none is repeated:
1
=
A
x
=
A
x
B
C
+ 2x−1
+ x+2
B
C
D
+ x−1
+ (x−1)
2 + (x−1)3
- Partial fractions:
x
(x−2)(x2 +1)(x2 +4)
A
x−2
=
- Completing square then substitution:
Bx+C
+ Dx+E
x2 +1
x2 +4
x−1
x−1
= (2x−1)
2 +2 ,
4x2 −4x+3
+
(4)Q(x) contains a repeated irreducible quadratic factor:
(5) Rationalizing substitutions u =
p
n
g(x):
√
x+4
x ,
let u =
let u = 2x − 1.
x+1
(x2 +1)3
√
=
Ax+B
x2 +1
+
Cx+D
(x2 +1)2
+
Ex+F
(x2 +1)3
x + 4.
(C) Radicals
(1)√Trigonometric Substitutions:
If a2 − x2 , let x = a sin θ, 1 − sin2 θ = cos2 θ.
If
If
√
√
(2)
a2 + x2 , let x = a tan θ, 1 + tan2 θ = sec2 θ.
x2 − a2 , let x = a sec θ, sec2 θ − 1 = tan2 θ.
p
p
√
√
n
g(x), let u = n g(x), un = g(x): 3 x + 4, let u = 3 x + 4, then u3 = x + 4.
Other Important Trigonometric Identities
1
sin A cos B = [sin(A − B) + sin(A + B)]
2
1
sin A sin B = [cos(A − B) − cos(A + B)]
2
1
cos A cos B = [cos(A − B) + cos(A + B)]
2
II Improper Integrals
proposition
R1
proposition
R∞
1
0 xp dx
1
is convergent if p < 1 and divergent if p ≥ 1.
1
xp dx
is convergent if p > 1 and divergent if p ≤ 1.
Comparison Theorem Suppose that f and g are continuous functions with f (x) ≥ g(x) ≥ 0
for x ≥ a.
(a) If
R∞
(b) If
R∞
a
a
f (x)dx is convergent, then
g(x)dx is divergent, then
R∞
a
R∞
a
g(x)dx is convergent.
f (x)dx is divergent.
III Taylor Polynomials and Taylor Series
1. The nth Taylor polynomial for f based at b is
n
X
1 (k)
1 (2)
1
Tn (x) =
f (b)(x − b)k = f (b) + f 0 (b)(x − b) +
f (b)(x − b)2 + ... + f (n) (b)(x − b)n
k!
2·1
n!
k=0
.
2
2. Taylor’s Inequality Suppose I is an interval containing b. If | f (n+1) (t) |≤ M for all t in
I, then
M
| f (x) − Tn (x) |≤
|x − b|n+1
(n + 1)!
for all x in I, where Tn is the nth Taylor polynomial for f based at b.
3. The Taylor’s Series for f based at b is defined to be
∞
X
1 (k)
lim Tn (x) =
f (b)(x − b)k
n→∞
k!
k=0
4. Important Series
x
e =
∞
X
xk
k=0
1
=
1−x
cos(x) =
k!
∞
X
k=0
∞
X
k=0
xk
x2k
(−1)
(2k)!
ln(1 − x) =
k
∞
X
k=1
1
− xk
k
3
sin(x) =
∞
X
(−1)k
k=0
tan−1 (x) =
x2k+1
(2k + 1)!
∞
X
(−1)k 2k+1
x
2k + 1
k=0