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APPM 1360
Exam 1 Study Guide
Spring 2016
Trapezoidal Rule
Integration by Parts
b
Z
Z
u dv = uv −
∆x
2
f (x) dx ≈ Tn =
Z
v du
a
[f (x0 ) + 2f (x1 ) + · · · + 2f (xn−1 ) + f (xn )]
where ∆x = (b − a)/n and xi = a + i∆x.
Numerical Integration Error Bounds
Trigonometric Identities
∗
∗
sin 2θ = 2 sin θ cos θ
1 + tan2 θ = sec2 θ
∗
cos 2θ = 2 cos2 θ − 1
1 + cot2 θ = csc2 θ
∗
cos 2θ = 1 − 2 sin2 θ
sin2 θ + cos2 θ = 1
Trigonometric Integrals
Z
tan x dx = ln|sec x| + C
Z
cot x dx = ln|sin x| + C
Z
sec x dx = ln|sec x + tan x| + C
Z
csc x dx = ln|csc x − cot x| + C
Trigonometric Substitutions
p
a2 − x2
p
a2 + x2
p
x2 − a2
Z
|ET | ≤
K(b − a)3
12n2
∗
|EM | ≤
K(b − a)3
24n2
where |f 00 | ≤ K.
Improper Integrals (infinite intervals or discontinuous integrands)
Z ∞
Z a
Z ∞
f (x) =
f (x) dx +
f (x) dx
1
1
x − sin 2x + C
2
4
Z
1
1
cos2 x dx = x + sin 2x + C
2
4
Z
dx
1
∗
−1 x
= tan
+C
2
2
x +a
a
a
Z
x
dx
∗
√
+C
= sin−1
a
a2 − x2
sin2 x dx =
−∞
∞
−∞
Z
a
Z
f (x) = lim
t→∞
a
t
Z
b
f (x) dx
a
Z
f (x) = lim
−∞
t→−∞
x = a sin θ
x = a tan θ
Convergent Integrals
Z ∞
1
dx (p > 1)
xp
Z1 ∞
e−x dx
x = a sec θ
Divergent Integrals
Z ∞
1
dx (p ≤ 1)
xp
1
0
Ax + B
(ax2 + bx + c)n
f (x) dx
t
Comparison Test
Suppose that f and g are continuous functions with f (x) ≥ g(x) ≥ 0
for x ≥ a.
R∞
R∞
1. If a f (x) dx is convergent, then a g(x) dx is convergent.
R∞
R∞
2. If a g(x) dx is divergent, then a f (x) dx is divergent.
Partial Fractions
A
(ax + b)n
b
∗
You need not memorize formulas marked with an asterisk.
Volume by Slicing
Z
b
A(x) dx
V =
a
Disk/Washer Method
Z b
V =
πr2 dx
Z
a
Z
V =
a
b
π R2 − r2 dx
or V =
a
b
πr2 dy
Z
or V =
b
π R2 − r2 dy
a
Note: This study guide is a summary of the material that has been covered
in this unit. It is not a complete list of all topics that may appear on the exam.