Formula Sheet for Math 175

Formula Sheet for Math 175
Integrals
Z
Z
tan x dx = ln | sec x| + C
cot x dx = − ln | csc x| + C
Z
Z
sec x dx = ln | sec x + tan x| + C
Z
csc x dx = − ln | csc x + cot x| + C
Z
sec2 x dx = tan x + C
Z
Z
csc x cot x dx = − csc x + C
sec x tan x dx = sec x + C
Z
csc2 x dx = − cot x + C
√
Z
x
dx
= sin−1
+C
a
a2 − x2
Trigonometric Substitution
p
a2 − x2 −→ x = a sin θ,
p
dx
1
−1 x
=
tan
+C
a2 + x2
a
a
a2 + x2 −→ x = a tan θ,
p
x2 − a2 −→ x = a sec θ
Trigonometric Identities
1 + tan2 θ = sec2 θ,
cos2 θ =
1 + cos 2θ
,
2
sin2 θ =
1 + cot2 θ = csc2 θ
1 − cos 2θ
,
2
sin 2θ = 2 sin θ cos θ
Series
f 00 (a)
f 000 (a)
(x − a)2 +
(x − a)3 + · · ·
2!
3!
k(k − 1)x2
k(k − 1)(k − 2)x3
(1 + x)k = 1 + kx +
+
+ · · · , −1 < x < 1
2!
3!
a
a + ar + ar2 + ar3 + · · · =
, −1 < r < 1
1−r
x3 x5 x7
x2 x4 x6
x2 x3 x4
sin x = x −
+
−
+ · · · , cos x = 1 −
+
−
+ · · · , ex = 1 + x +
+
+
+···
3!
5!
7!
2!
4!
6!
2!
3!
4!
f (x) = f (a) + f 0 (a)(x − a) +
Differential Equations
dy
+ P (x)y = Q(x)
dx
R
Integrating factor: e
P (x) dx
ay 00 (x) + by 0 (x) + cy(x) = 0
y = c1 em1 x + c2 em2 x
y = (c1 + c2 x)emx
y = eαx (c1 cos(βx) + c2 sin(βx))
Fourier Series
If f (x) has period T = 2L, then
f (x) = a0 +
∞
X
ak cos
k=1
1
a0 =
2L
Z
L
f (x) dx,
−L
1
ak =
L
Z
L
f (x) cos
−L
kπx
L
kπx
L
dx,
+
∞
X
bk sin
k=1
1
bk =
L
Z
kπx
L
L
f (x) sin
−L
kπx
L
dx,
k = 1, 2, 3 . . .
f (t)
L {f (t)} = F (s)
f (t)
1
1
s
eat
1
s−a
eat tn
n!
1
(s − a)n+1
n!
tn
sn+1
L {f (t)} = F (s)
sin ωt
ω
s2 + ω 2
cos ωt
s
s2 + ω 2
eat sin ωt
ω
(s − a)2 + ω 2
eat cos ωt
s−a
(s − a)2 + ω 2
t sin ωt
2ωs
(s2 + ω 2 )2
t cos ωt
s2 − ω 2
(s2 + ω 2 )2
sin ωt − ωt cos ωt
2ω 3
1
(s2 + ω 2 )2
U(t − a)
e−as
s
δ(t)
1
δ(t − a)
e−as
L {αf (t) + βg(t)} = αL {f (t)} + βL {g(t)}
L {eat f (t)} = F (s − a)
L {tn f (t)} = (−1)n
L −1 {e−as F (s)} = f (t − a) U(t − a)
L {f 0 (t)} = sF (s) − f (0)
dn
F (s)
dsn
L {f (t) U(t − a)} = e−as L {f (t + a)}
L {f 00 (t)} = s2 F (s) − sf (0) − f 0 (0)
L {f (n) (t)} = sn F (s) − sn−1 f (0) − · · · − f (n−1) (0)
Z
f (t) ∗ g(t) =
t
f (θ)g(t − θ) dθ =⇒ L {f (t) ∗ g(t)} = F (s)G(s)
0
L
Z
t
f (θ) dθ
=
0
f (t) has period T =⇒ L {f (t)} =
F (s)
s
1
1 − e−sT
Z
0
T
e−st f (t) dt