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Integral Calculus Formula Sheet
Derivative Rules:
d c dx
(^) n n^1
d x nx dx
^
sin^ cos
d x x dx
sec^ sec^ tan
d x x x dx
2 tan sec
d x x dx
cos^ sin
d x x dx
csc^ csc^ cot
d x x x dx
2 cot csc
d x x dx
ln
d (^) x x a a a dx
d (^) x x e e dx
d d cf x c f x dx dx
d d d f x g x f x g x dx dx dx
f^ g^ f^ g^ f^ g ^ 2
f f g fg
g g
^
^ ^ ^ ^ ^ ^ ^
d f g x f g x g x dx
Properties of Integrals:
kf u du ( ) k f u du ( ) ^
f ( ) u g u ( ) du f u du ( ) g u du ( )
a
a
f^ x dx^ ( )^ ( )
b a
a b
f^ x dx^ f^ x dx
c b c
a a b
f^ x dx^ ^ f^ x dx^ f^ x dx
b
ave a
f f x dx b a
0
a a
a
f x dx f x dx
if^ f(x)^ is^ even^ ( )^0
a
a
f x dx
if^ f(x)^ is^ odd
( )
( )
b^ f^ b
a f a
g f x f x dx g u du udv^ ^ uv^ vdu
Integration Rules:
^ du^ ^ u^ C 1
n n u u du C n
ln
du u C u
^
u u e du^ ^ e^ C
1
ln
u u a du a C a
^ sin^ u du^ ^ cos u^ C
cos^ u du^ ^ sin u^ C 2 sec u du tan u C 2 csc^ u^ ^ cot u^ C
csc^ u^ cot^ u du^ ^ ^ csc u^ C
sec^ u^ tan^ u du^ ^ sec u^ C
2 2
arctan
du u C a u a a
2 2
arcsin
du u C a u a
^
2 2
sec
du u arc C u u a a^ a
Fundamental Theorem of Calculus:
x
a
d
F x f t dt f x
dx
where f t is a continuous function on [ a, x ].
^ ^ ^ ^ ^ ^
b
a
f x dx F b F a , where F(x) is any antiderivative of f(x).
Riemann Sums:
1 1
n n
i i i i
ca c a
1 1 1
n n n
i i i i i i i
a b a b
^ ^
1
( ) lim ( )
b (^) n
n a i
f x dx f a i x x
n
b a x
1
n
i
n
1
( 1)
2
n
i
n n i
2
1
( 1)(2 1)
6
n
i
n n n i
2 3
1
( 1)
2
n
i
n n i
height of th rectangle^ width of th rectangle
i
i^ i
Right Endpoint Rule:
n
i
n
ba n
b a
n
i
f a i x x f a i
1
( ) ( )
1
Left Endpoint Rule:
( ) ( )
1 1
n n b a b a n n i i
f a i x x f a i
^ ^ ^ ^ ^ ^
Midpoint Rule:
( 1) ( ) ( 1) ( ) 2 2 1 1
n n i i b a i i b a n n i i
f a x x f a
^ ^ ^ ^
Net Change:
Displacement: ( )
b
a
v x dx Distance^ Traveled:^ ( )
b
a
v x^ dx
0
t
s t s v x dx
0
t
Q t Q Q x dx
Trig Formulas:
(^2 ) sin ( ) x 21 cos(2 ) x
sin tan cos
x x x
sec cos
x x
cos( x ) cos( ) x sin ( )^2 x cos ( )^2 x 1
(^2 ) cos ( ) x 2 1 cos(2 ) x
cos cot sin
x x x
csc sin
x x
sin( x ) sin( ) x tan ( )^2 x 1 sec ( )^2 x
Geometry Fomulas:
Area of a Square: 2 A s
Area of a Triangle: 1 A 2 bh
Area of an Equilateral Trangle: 3 2 A 4 s
Area of a Circle: 2 A r
Area of a Rectangle:
A bh
Integration by Parts:
Knowing which function to call u and which to call dv takes some practice. Here is a general guide:
u I nverse Trig Function (
1 sin x , arccos x ,
etc )
L ogarithmic Functions ( log 3 , ln( x x 1),etc )
A lgebraic Functions (
3 x , x 5,1/ x ,etc )
T rig Functions ( sin(5 ), tan( ), x x etc )
dv E xponential Functions (
3 3 ,5 ,
x x e etc )
Functions that appear at the top of the list are more like to be u, functions at the bottom of the list are more like to be dv.
Trig Integrals:
Integrals involving sin(x) and cos(x): Integrals involving sec(x) and tan(x):
- If the power of the sine is odd and positive:
Goal: u cos x i. Save a du sin( ) x dx
ii. Convert the remaining factors to
cos( ) x (using
2 2 sin x 1 cos x .)
- If the power of sec( ) x is even and positive:
Goal: u tan x
i. Save a
2 du sec ( ) x dx
ii. Convert the remaining factors to
tan( ) x (using
2 2 sec x 1 tan x .)
- If the power of the cosine is odd and positive:
Goal: u sin x i. Save a du cos( ) x dx
ii. Convert the remaining factors to
sin( ) x (using
2 2 cos x 1 sin x .)
- If the power of tan( ) x is odd and positive:
Goal: u sec( ) x
i. Save a du sec( ) tan( ) x x dx
ii. Convert the remaining factors to
sec( ) x (using
2 2 sec x 1 tan x .)
- If both sin( ) x and cos( ) x have even powers:
Use the half angle identities:
i.
(^2 ) sin ( ) x 2 1 cos(2 ) x
ii.
(^2 ) cos ( ) x 2 1 cos(2 ) x
If there are no sec(x) factors and the power of
tan(x) is even and positive, use
2 2 sec x 1 tan x
to convert one
2 tan x to
2 sec x
Rules for sec(x) and tan(x) also work for csc(x) and cot(x) with appropriate negative signs If nothing else works, convert everything to sines and cosines.
Trig Substitution:
Expression Substitution Domain Simplification
2 2
a u u^ ^ a sin^ 2 2
a^2^ u^2 a cos
2 2
a u u^ ^ a tan^ 2 2
a^2 u^2 a sec
2 2
u a u^ ^ a sec^ ^0 ,^2
u^2 a^2^ a tan
Partial Fractions:
Linear factors: Irreducible quadratic factors:
2 1 1 1 1 1 1
( ) ... ( ) ( ) ( ) ( ) ( )
m m m
P x A B Y Z
x r x r x r x r x r
(^) 2 2 2 2 2 1 2 1 1 1 1 1
( ) ... ( ) ( ) ( ) ( ) ( )
m m m
P x Ax B Cx D Wx X Yx Z
x r x r x r x r x r
If the fraction has multiple factors in the denominator, we just add the decompositions.
