Calculus Formulas
n +1
∫
d n
x
x = nx n −1 and x n dx =
+c
dx
n +1
d ⎡ f (x ) ⎤ g (x ) ⋅ f ' (x ) − f (x ) ⋅ g ' (x )
Quotient Rule:
⎢
⎥=
dx ⎣ g (x ) ⎦
[g (x )]2
d
Chain Rule:
( f o g )(x ) = f ' [g (x )] ⋅ g ' (x )
dx
Power Rules:
Product Rule:
Reciprocal Rule:
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec 2 x
dx
d
cot x = − csc 2 x
dx
d
sec x = sec x ⋅ tan x
dx
d
csc x = − csc x ⋅ cot x
dx
Derivative
d
1
−1
sin x =
dx
1− x2
−1
d
cos −1 x =
dx
1− x2
∫ sin x dx = − cos x + c
∫ cos x dx = sin x + c
∫ tan x dx = ln sec x + c
∫ sec x dx = tan x + c
∫ cot x dx = ln sin x + c
∫ csc x dx = − cot x + c
∫ sec x dx = ln sec x + tan x + c
∫ sec x ⋅ tan x dx = sec x + c
∫ csc x dx = ln csc x − cot x + c
∫ csc x ⋅ cot x dx = − csc x + c
2
2
1
d
tan −1 x =
dx
1 + x2
−1
d
cot −1 x =
dx
1+ x2
d
1
sec −1 x =
dx
x x2 −1
d
−1
csc −1 x =
dx
x x2 −1
sin 2 x = 2 sin x cos x
cos(x + y ) = cos x cos y − sin x sin y
Exponential Functions
Integral
( )
∫e
( )
∫
d x
e = ex
dx
d x
b = (ln b )b x
dx
x
dx = e x + c
b x dx =
bx
+c
ln b
Definition of Log base b: log b N = x ⇔ b x = N
( )
( )
⎧⎪ln e x = x
Identities: ⎨
⎪⎩log b b x = x
e ln x = x
b
log b x
=x
Integral
∫
1
a −u
∫a
∫u
2
dx = sin −1
2
2
1
dx =
+ u2
1
u2 − a2
dx =
1
u
sec −1 + c
a
a
1 + cos 2 x
2
1
cos
2x
−
sin 2 x =
2
sin (x + y ) = sin x cos y + cos x sin y
Logarithmic Functions
Derivative
d
(ln x ) = 1
dx
x
d
(log b x ) = ( 1 )
dx
ln b x
Integral
∫ x dx = ln x + c
1
Change of Base Formula: log b x =
ln e = log 10 = log b b = 1
ln 1 = log 1 = log b 1 = 0
u
+c
a
u
1
tan −1 + c
a
a
cos 2 x =
cos 2 x = cos 2 x − sin 2 x
Derivative
∫ u dv = uv − ∫ v du
Inverse Trigonometric Functions
Integral
⎧ 2
2
⎪sin x + cos x = 1
⎪
⎪
Identities: ⎨1 + cot 2 x = csc 2 x
⎪
⎪tan 2 x + 1 = sec 2 x
⎪
⎩
d ⎡ 1 ⎤ − g ' (x )
⎢
⎥=
dx ⎣ g (x ) ⎦ [g (x )]2
Integration-by-Parts:
Trigonometric Functions
Derivative
d
[ f (x ) ⋅ g (x )] = f (x ) ⋅ g ' (x ) + f ' (x ) ⋅ g (x )
dx
ln x log x
=
ln b log b
Infinite Series: Definitions & Tests
⎧
⎪ a n = a1 + a 2 + a 3 + ... (Infinite Series )
⎪ n =1
⎪
n
⎪
Series: ⎨s n =
a i = a1 + a 2 + ... + a n (nth Partial Sum )
⎪
i =1
⎪
∞
⎪if lim s = s where s ∈ ℜ then
a n = s (Infinite Sum )
⎪ n →∞ n
n =1
⎩
∞
∑
1.
∑
∑
⎧ a
, if r < 1
⎪
ar n = a + ar + ar 2 + ar 3 + ... = ⎨1 − r
⎪diverges, if r ≥ 1
n =0
⎩
∞
2.
Geometric Series:
3.
P-Series:
∞
∑n
n =1
1
p
∑
⎧converges, if p > 1
⇒⎨
⎩diverges, if p ≤ 1 if p = 1, the series is called the harmonic series.
∞
⎧
a n diverges
⎪⎪if lim a n ≠ 0, then
Quick Divergence Test: Given
a n ⇒ ⎨ n →∞
n =1
⎪if lim a = 0, then No Conclusion! Do another test!
n =1
⎪⎩ n→∞ n
∞
⎧ ∞
⎪if a n dn converges then
a n converges
∞
⎪⎪ c
n =c
Integral Test: Given
a n , a n > 0, a n decreasing ⇒ ⎨
∞
∞
⎪
n =c
if
a
dn
diverges
then
a n diverges
⎪
n
⎪⎩ c
n =c
∑
∞
4.
5.
∑
∑
∫
∑
∑
∫
⎧∞
⎪ a n converges, when p < 1,
⎪ n =c
⎪∞
⎪
= p then ⎨ a n diverges, when p > 1,
⎪ n =c
⎪ No Conclusion, when p = 1
⎪
⎪
⎩
∑
∞
6.
Ratio Test: Given
∑a
n,
a n > 0 ⇒ if lim
n →∞
n =c
a n +1
an
∑
⎧∞
⎪ a n converges, when p < 1,
⎪ n =c
⎪∞
∞
1
⎪
n
(
)
a n , a n > 0 ⇒ if lim a n = lim a n n = p then ⎨ a n diverges, when p > 1,
Root Test: Given
n →∞
n→∞
⎪ n =c
n =c
⎪ No Conclusion, when p = 1
⎪
⎪
⎩
an
⎧
∞
∞
= p, p > 0, p finite
⎪ if nlim
→
∞
bn
a
b
,
a
0
,
b
0
and
>
>
⇒
Limit Comparison Test:
⎨
n
n
n
n
⎪then both series converge or both diverge
n =c
n =c
⎩
∑
7.
8.
∑
∑
∑
∞
9.
Comparison Test:
∑
∑
∞
a n and
n =c
∑b
n =c
n
⎧ if bn converges then a n converges,
, a n ≥ 0, bn ≥ 0, a n ≤ bn ⇒ ⎨
⎩if a n diverges then bn diverges
∞
∞
10.
Alternating Series Test: Given
∑ (− 1) a
n
n =c
n
, if a n > 0, a n +1 < a n , lim a n = 0, then
n →∞
∑ (− 1) a
n
n =c
n
converges