Trigonometric Identities
Trigonometric Identities
sin2(x) = 1 - cos(2x) 2
cos2(x) = 1 + cos(2x) 2
Reduction Formulas
sinn(x)dx
=
sinn-1(x) -
cos(x)
+
n
-
1
sinn-2(x)dx
n
n
cosn(x)dx = cosn-1(x) sin(x) + n - 1 cosn-2(x)dx
n
n
tann(x)dx = tann-1(x) - tann-2(x)dx n-1
cotn(x)dx
=
cotn-1(x) -
-
cotn-2(x)dx
n-1
secn(x)dx = tan(x) secn-2(x) + n - 2 secn-2(x)dx
n-1
n-1
cscn(x)dx
=
cot(x) -
cscn-2(x)
+
n
-
2
cscn-2(x)dx
n-1
n-1
Other Integration Formulas
dx
1
x
x2
+
a
=
a
arctan
a
+C
Important Power Series
(for a > 0)
1
= xk = 1 + x + x2 + x3 + . . .
1-x
k=0
ex =
xk
x2 x3 = 1+x+ + +...
k!
26
k=0
(-1)kx2k+1
x3 x5
x7
sin(x) =
=x- + -
+...
(2k + 1)!
6 120 5040
k=0
(-1)kx2k
x2 x4 x6
cos(x) =
= 1- + - +...
(2k)!
2 24 720
k=0
xk
x2 x3 x4
ln(1 - x) =
= x + + + + . . . arctan(x)
k
234
k=1
(-1)kx2k+1
x3 x5 x7
=
= x- + - +...
2k + 1
357
k=0
Differential Eqauations
dy = k(y - b)
dt
dy
y
= ky 1 -
dt
A
y(t) = Cekt + b A
y(t) = 1 - Ce-kt (nonequilibrium)
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