1
Q
integral sec x tan x dx=
A
sec x+C
2
Q
integral 0 dx=
A
C
3
Q
integral 1 dx=
A
x+C
4
Q
integral e x dx=
A
e x+C
5
Q
integral 1/x dx=
A
ln x+C
6
Q
integral n^xdx
A
n^x / ln x+C
7
Q
integral cos x dx=
A
sin x+C
8
Q
integral sin x dx=
A
_cos x+C
9
Q
integral sec^2 x dx=
A
tan x+C
10
Q
integral csc^2 x dx=
A
_cot x+C
11
Q
integral sec x tan x dx=
A
sec x+C
12
Q
integral csc x cot x dx=
A
_csc x+C
13
Q
d/dx ln x
A
1/x
14
Q
d/dx sinh u=
A
cosh u
15
Q
d/dx cosh u=
A
sinh u
16
Q
d/dx tanh u=
A
Sech^2 u
17
Q
integral sinhu du=
A
cosh u + C
18
Q
integral cosh u du=
A
sinh u + C
19
Q
integral sech^2 u du=
A
tanh u + C
20
Q
integral csch^2 u du=
A
-coth u + C
21
Q
integral sech u tanh u du=
A
-sech u + C
22
Q
integral csch u coth u du=
A
-csch u + C
23
Q
d/dx coth u=
A
-csch^2 u
24
Q
d/dx sech u =
A
-sech u tanh u
25
d/dx csch u =
-csch u coth u
26
d/dx of ln(x)=
1/x
27
d/dx ln ( f(x) ) =
1/f(x) (f"(x))
28
d/dx loga(X)=
1/xlna
29
d/dx log a (f(x))=
1/(f(x) lna )
30
product rule g(f(x))=
g'f x f'g
31
Chain rule
getting the derivative of out side and leave inside alone and then multiplying by the derivative of inside
32
you an write the square root of X as X to the power 1/2, how about ln of square root of X
ln of square root of x we can rewrite as ln of x to the power 1/2 and then rewrite as 1/2lnx
33
ln (abcd)=
(lna)(lnb)(lnc)(lnd)
34
Log a (x)=y
a^y=x
35
loga (xy)=
loga (x) + loga (y)
36
loga (x/y)=
loga (x) - loga (y)
37
loga (a^x)=
x
38
lnx=e then e^lnx=e^e
x=e^e
39
change base formula
loga (b) = ln(b) / ln(a) remember base goes to the basement (bottom)
40
d/dx a^x=
a^x lna
41
d/dx a^x=
a^x lna
42
d/dx a^f(x)
a^f(x) lna f'(x)
43
d/dx of ln(x)=
1/x
44
d/dx ln ( f(x) ) =
1/f(x) (f"(x))
45
d/dx loga(X)=
1/xlna
46
d/dx log a (f(x))=
1/(f(x) lna )
47
product rule g(f(x))=
g'f x f'g
48
Chain rule
getting the derivative of out side and leave inside alone and then multiplying by the derivative of inside
49
you an write the square root of X as X to the power 1/2, how about ln of square root of X
ln of square root of x we can rewrite as ln of x to the power 1/2 and then rewrite as 1/2lnx
50
ln (abcd)=
(lna)(lnb)(lnc)(lnd)
51
Log a (x)=y
a^y=x
52
loga (xy)=
loga (x) + loga (y)
53
loga (x/y)=
loga (x) - loga (y)
54
loga (a^x)=
x
55
lnx=e then e^lnx=e^e
x=e^e
56
change base formula
loga (b) = ln(b) / ln(a) remember base goes to the basement (bottom)
57
d/dx a^x=
a^x lna
58
d/dx a^x=
a^x lna
59
d/dx a^f(x)
a^f(x) lna f'(x)
60
Half life
Ln2/k or Ln0.5/k
61
Decay and growth
A(t)=A0e^rt Y=Y0e^kt R is the interest or percent T is number of years A0 is initial value
62
Temp formula
T-Ts=(T0-Ts)e^-kt T is the current temp or the one we want to get to and Ts is the room temp and T0 is the initial temp
63
Hyperbolic sinh
(e^x - e^-x) / 2
64
Hyperbolic cosh
(e^x + e^-x) / 2
65
Cosh^2x - sinh^2x=
1
66
Sinh2x
2sinhx
67
what do we use if we have squer root of (a2 - X2)
then we put X= a sin
68
what do we use if we have squer root of (a2 + X2)
then we put X=a tan Ø
69
what do we use if we have squer root of (X2 - a2)
then we put X = a Sec
70
sin2X=
2 sinX cosX
71
integral of Sec X dx=
ln I secx+tanx I + c
72
integral of Csc x dx=
ln I csc x - cot x I + c
73
Sec2X - 1
tan2X
74
integral of (tan x) dx=
ln I sec x I + c
75
integral of Sec X dx=
ln I sec x + tan x I + c
76
Cos 2X =
1/2 ( 1 + cos (2x))
77
sin2X=
1/2 ( 1 - cos (2x))
78
1+ tan2X =
sec2X
79
integral of sin (mx) Cos (nx) dx=
we have to use Sine A Cos B =
1/2 [sin (A-B) + sin ( A+B)]
80
integral of sin (mx) sin (nx) dx=
we have to use Sine A sin B =
1/2 [cos (A-B) - cos ( A+B)]
81
integral of cos (mx) Cos (nx) dx=
## Footnote
we have to use cos A Cos B =
1/2 [cos (A-B) + cos (A+B)]
Calculus Formulas
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