Derivatives and Integral Formulas Flashcards
(64 cards)
1
Q
d/dx(sinax)
A
acos(ax)
2
Q
d/dx(cosax)
A
-asin(ax)
3
Q
d/dx(tanax)
A
asec^2(ax)
4
Q
d/dx(cotx)
A
acsc^2(ax)
5
Q
d/dx(secax)
A
asecaxtanax
6
Q
d/dx(cscax)
A
-acscaxcotax
7
Q
d/dx(e^ax)
A
ae^ax
8
Q
d/dx(b^x)
A
b^xlnb
9
Q
d/dx(ln|x|)
A
1/x
10
Q
Power Rule: d/dx(x^n)
A
nx^n-1
11
Q
Product Rule: d/dx(f(x)*g(x))
A
f’(x)g(x)+g’(x)f(x)
12
Q
Quotient Rule: d/dx(f(x)/g(x))
A
f’(x)g(x)+g’(x)f(x)/(g(x))^2
13
Q
∫ cosaxdx
A
1/a(sinax)+C
14
Q
∫ sinaxdx
A
-1/a(cosax)+C
15
Q
∫ sec^(2)axdx
A
1/a(tanax)+C
16
Q
∫ csc^(2)axdx
A
-1/a(cotax)+C
17
Q
∫ secaxtanaxdx
A
1/a(secax)+C
18
Q
∫ cscaxcotaxdx
A
-1/a(cscax)+C
19
Q
∫ e^axdx
A
1/ae^ax+C
20
Q
∫ b^xdx
A
1/lnb(b^x)+C
21
Q
∫ (1/x)dx
A
ln|x|+C
22
Q
∫ (1/a^2+x^2)
A
1/a(tan^-1(x/a))+C
23
Q
∫ (1/xsqrt(x^2-a^2))
A
1/a(sec^-1|x/a|+C
24
Q
∫ (1/sqrt(a^2-x^2))
A
sin-1(x/a)+C
25
d/dx(sin^-1x)
1/sqrt(1-x^2)
26
d/dx(cos^-1x)
-1/sqrt(1-x^2)
27
d/dx(tan^-1x)
1/1+x^2
28
d/dx(cot^-1x)
-1/1+x^2
29
d/dx(sec^-1x)
1/xsqrt(x^2-1)
30
d/dx(csc^-1x)
-1/xsqrt(x^2-1)
31
rec sec(x)
1/(cosx)
32
rec csc(x)
1/(sinx)
33
rec cot(x)
1/(tanx)
34
rec cos(x)
1/(secx)
35
rec sin(x)
1/(cscx)
36
rec tan(x)
1/(cotx)
37
tan(x)
sinx/cosx
38
cot(x)
cosx/sinx
39
sin^2x+cos^2x
1
40
1+tan^2x
sec^2x
41
1+cot^2x
csc^2x
42
sin^2(A)
1/2(1-cos(2A))
43
cos^2(A)
1/2(1+cos(2A))
44
sin(2A)
2sin(A)cos(A)
45
cos(2A)
2cos^2(A)-1,1-2sin^2(A),cos^2(A)-sin^2(A)
46
∫
integral sign
47
∫f(x)dx what is f(x)
integrand
48
∫tan(ax)dx
-1/aln|cos(ax)|+C
49
∫cot(ax)dx
1/aln|sin(ax)|+C
50
∫sec(ax)dx
1/aln|sec(ax)+tan(ax)|+C
51
∫csc(ax)dx
1/aln|csc(ax)+cot(ax)|+C
52
∫udv
uv-∫vdu
53
acronym for integration by parts
LIATE, log, inverse trigonometric, algebraic, trignometric, exponential
54
sin^m(x)cos^n(x) if power of sin is odd
save one sine factor with the dx, pythagorean identity, u-substitution
55
sin^m(x)cos^n(x) if power of cosine odd
save one cosine factor with the dx, pythagorean identity, u-substitution
56
sin^m(x)cos^n(x) all powers even
power reducing formula, simplify
57
sec^m(x)tan^n(x) if the power of secant is even
save a factor of sec^2(x) with dx, Use sec^2(x)=1+tan^2(x), use u-substitution
58
sec^m(x)tan^n(x) if the power of tangent is odd
save a factor of sec(x)tan(x) with dx, use tan^2(x)=sec^2(x)-1,use u-substitution
59
∫tan^n(x)dx
(1/n-1)tan^n-1(x)-∫tan^n-2(x)dx
60
∫sec^n(x)dx
(1/n-1)tan(x)sec^n-2(x)+(n-2/n-1)∫sec^n-2(x)dx
61
to use trigonometric substitution which 3 expressions must be included
a^2-x^2,x^2-a^2,x^2+a^2
62
a^2-x^2
x=asin(theta), dx=acos(theta)dx
63
x^2-a^2
x=asec(theta), dx=asec(theta)tan(theta)dtheta
64
a^2+x^2
x=atan(theta), dx=asec^2(theta)dtheta
