CALCULUS CHEAT SHEET Derivative of Trigonometric Functions d u

e=
u du
e dx
dx
d du
DERIVATION RULES dx
sin u = cos u dx
Derivative of General Exponential Functions
d du d u u du
Derivative of Algebraic Functions dx
cos u =−sin u dx a =a ln a dx
dx
d du
d n n−1
tan u = sec2 u
u = nu dx dx
dx d du INTEGRATION FORMULAS
d cot u = –csc 2 u
C =0 dx dx
dx d du Integration
d du dx
sec u = sec u tan u dx
dx
cu= c dx d du
csc u = –csc u cot u un + 1
n
d du dv
( u + v )= dx + dx dx dx ∫ u du = n + 1 + c
dx
d
√ u=
1 du Derivative of Inverse Trigonometric Functions ∫ du=u +C
dx 2 √ u dx
d dv du d 1 du
∫ a du=a∫ du +C
dx
( u ⋅ v)= u +v
dx dx
arcsin u =
d dw dv du
dx √ 1 − u2 dx The Definite Integral
( uvw)=uv +uw +vw d −1 du
dx dx dx dx arccos u =
∫a f ( x) dx =F ( x ) |ba=F ( b)−F ( a )
b
du dv
v −u
dx √1 − u2 dx
d
dx ()
u
v
= dx 2 dx
v
d
dx
arctan u =
1 du
1 + u2 dx
b
∫a k dx=k ( b − a )
d n n− 1 du d −1 du b a
dx
u = nu dx
arccot u = ∫ a f (x) dx=−∫ b f (x) dx
dx 1 + u2 dx
d 1 du
Derivative of Logarithmic Functions arcsec u = Integral s involving Logarithmic Functions
dx |u| √u2 − 1 dx
d 1 du d −1 du 1
lnu= arccsc u = ∫ u du=ln | u | +C
dx u dx dx |u| √u2 − 1 dx
d 1 du
dx
log a u=
u ln a dx
Derivative of Exponential Functions
Integral of a Exponential Function

∫ eu du=eu +C
Integral involving Trigonometric du 1 u
Functions ∫ = arcsec +C
√ u −a
2 2 a a
∫ cos u du=sin u +C
∫ sin u du=−cos u +C Integration by Parts
CALCULUS
∫ sec u tan u du=sec u +C CHEAT SHEET
∫ csc u cot u du=−csc u +C ∫ u dv=uv −¿ ∫ v du +C ¿
∫ sec2 u du=tan u +C
∫ csc2 u du=−cot u +C
∫ tan u du=−ln|cos u|+C|sec u|
∫ cot u du=ln|sin u| +C
Wallis’ Formula
∫ sec u du=ln|sec u+tan u| +C
∫ csc u du=ln|csc u−cot u| +C π /2 m n
∫0
sin x cos x dx
Integral involving Inverse Trigonometric
Functions =
[ ( m−1 )( m−3) ⋯2 or 1][ ( n−1 )( n−3 ) ⋯2 or 1 ] ⋅α
( m+n )( m+n−2) ⋯2 or 1 Jerimar A. Beboso
du u
∫ 2 2 =arcsin a +C π STEM D
√ a −u Where:
α=
2 , if m and n are both even
du 1 u
∫ 2
a +u 2
=
a
arctan
a
+C α = 1, if otherwise