Integral Calculus Formula Sheet
Integral Calculus Formula Sheet
Derivative Rules:
d dx
c
0
d sin x cos x
dx
d sec x sec x tan x
dx
d tan x sec2 x
dx
d dx
cf
x
c
d dx
f
x
f g f g f g
d
dx
xn
nx n 1
d cos x sin x
dx
d csc x csc x cot x
dx
d cot x csc2 x
dx
d dx
f
x
g
x
d dx
f
x
d dx
g
x
f f g fg
g
g2
d ax ax ln a
dx
d ex ex
dx
d dx
f
g
x
f
g
x
g
x
Properties of Integrals:
kf (u)du k f (u)du
a
f (x)dx 0
a
c
b
c
f (x)dx f (x)dx f (x)dx
a
a
b
a
a
f (x)dx 2 f (x)dx if f(x) is even
a
0
b
f (b)
g( f (x)) f (x)dx g(u)du
a
f (a)
f (u) g(u)du f (u)du g(u)du
b
a
f (x)dx f (x)dx
a
b
fave
1 ba
b a
f
(x)dx
a
f (x)dx 0 if f(x) is odd
a
udv uv vdu
Integration Rules:
du u C
undu un1 C
n 1
du u
ln
u
C
eudu eu C
audu 1 au C ln a
sin u du cos u C cos u du sin u C sec2 u du tan u C csc2 u cot u C csc u cot u du csc u C sec u tan u du sec u C
du a2 u2
1 a
arctan
u a
C
du a2 u2
arcsin
u a
C
u
du u2 a2
1 a
arc
sec
u a
C
Fundamental Theorem of Calculus:
F ' x d x f t dt f x where f t is a continuous function on [a, x]. dx a
b f x dx F b F a , where F(x) is any antiderivative of f(x). a
Riemann Sums:
n
n
cai c ai
i 1
i 1
n
n
n
ai bi ai bi
i1
i 1
i 1
n
1 n
i 1
n
i
n(n
1)
i 1
2
n i2 n(n 1)(2n 1)
i 1
6
n
i 1
i3
n(n 1) 2 2
b
n
f (x)dx lim f (a ix)x
a
n i1
x b a n
height of ith rectangle width of ith rectangle
i
Right Endpoint Rule:
n
n
f (a ix)(x)
(
(ba) n
)
f
(a
i
(ba) n
)
i 1
i 1
Left Endpoint Rule:
n
n
f (a (i 1)x)(x)
(
(ba) n
)
f
(a
(i
1)
(ba) n
)
i 1
i 1
Midpoint Rule:
n
n
f (a
( i 1) i 2
x)(x)
(
(ba) n
)
f
(a
( i 1) i 2
) ( b a ) n
i 1
i 1
Net Change:
b
b
t
t
Displacement: v(x)dx Distance Traveled: v(x) dx s(t) s(0) v(x)dx Q(t) Q(0) Q(x)dx
a
a
0
0
Trig Formulas:
sin2 (x)
1 2
1
cos(2x)
cos2 (x)
1 2
1
cos(2x)
tan x sin x cos x
cot x cos x sin x
sec x 1 cos x
csc x 1 sin x
cos(x) cos(x) sin2 (x) cos2 (x) 1 sin(x) sin(x) tan2 (x) 1 sec2 (x)
Geometry Fomulas:
Area of a Square: A s2
Area of a Triangle:
A
1 2
bh
Area of an
Equilateral Trangle:
A
3 4
s2
Area of a Circle:
A r2
Area of a Rectangle:
A bh
Areas and Volumes:
Area in terms of x (vertical rectangles):
b
(top bottom)dx
a
General Volumes by Slicing: Given: Base and shape of Cross-sections
b
V A(x)dx if slices are vertical a d
V A( y)dy if slices are horizontal c
Washer Method: For volumes of revolution not laying on the axis with slices perpendicular to the axis
b
V R(x)2 r(x)2 dx if slices are vertical a d
V R( y)2 r( y)2 dy if slices are horizontal c
Area in terms of y (horizontal rectangles):
d
(right left)dy
c
Disk Method: For volumes of revolution laying on the axis with slices perpendicular to the axis
b
V R(x)2 dx if slices are vertical a
d
V R( y)2 dy if slices are horizontal c
Shell Method: For volumes of revolution with slices parallel to the axis
b
V 2 rhdx if slices are vertical a
d
V 2 rhdy if slices are horizontal c
Physical Applications: Physics Formulas
Mass: Mass = Density * Volume (for 3-D objects) Mass = Density * Area (for 2-D objects) Mass = Density * Length (for 1-D objects)
Work: Work = Force * Distance Work = Mass * Gravity * Distance Work = Volume * Density * Gravity * Distance
Force/Pressure: Force = Pressure * Area Pressure = Density * Gravity * Depth
Associated Calculus Problems
Mass of a one-dimensional object with variable linear
density:
b
b
Mass (linear density) dx (x)dx
a
distance a
Work to stretch or compress a spring (force varies):
b
b
b
Work ( force)dx F(x)dx kx dx
a
a
a Hooke's Law
for springs
Work to lift liquid:
d
Work (gravity)(density)(distance) (areaofa slice)dy
c
volume
d
W 9.8* * A( y) *(a y)dy (in metric)
c
Force of water pressure on a vertical surface:
d
Force (gravity)(density)(depth) (width)dy
c
area
d
F 9.8* *(a y) * w( y)dy (in metric)
c
Integration by Parts:
Knowing which function to call u and which to call dv takes some practice. Here is a general guide:
u
Inverse Trig Function ( sin1 x, arccos x, etc )
Logarithmic Functions ( log 3x, ln(x 1), etc )
Algebraic Functions
( x3, x 5,1/ x,etc )
Trig Functions
( sin(5x), tan(x),etc )
dv
Exponential Functions ( e3x , 53x , 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): 1. 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 sin2 x 1 cos2 x .)
Integrals involving sec(x) and tan(x): 1. If the power of sec(x) is even and positive:
Goal: u tan x i. Save a du sec2 (x)dx ii. Convert the remaining factors to tan(x) (using sec2 x 1 tan2 x .)
2. 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 cos2 x 1 sin2 x .)
2. If the power of tan(x) is odd and positive: Goal: u sec(x) i. Save a du sec(x) tan(x)dx ii. Convert the remaining factors to sec(x) (using sec2 x 1 tan2 x .)
3. If both sin(x) and cos(x) have even powers:
If there are no sec(x) factors and the power of
Use the half angle identities:
i. sin2 (x) 1 1 cos(2x) 2
tan(x) is even and positive, use sec2 x 1 tan2 x to convert one tan2 x to sec2 x
ii. cos2 (x) 1 1 cos(2x) 2
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
a2 u2
a2 u2 u2 a2
Substitution
u a sin u a tan u a sec
Domain
2
2
2
2
0 , 2
Simplification
a2 u2 a cos a2 u2 a sec u2 a2 a tan
Partial Fractions:
Linear factors:
Irreducible quadratic factors:
P(x) (x r1)m
A (x r1)
B (x r1)2
...
Y (x r1)m1
Z (x r1)m
P(x) (x2 r1)m
Ax B (x2 r1)
Cx D (x2 r1)2
...
Wx X (x2 r1)m1
Yx Z (x2 r1)m
If the fraction has multiple factors in the denominator, we just add the decompositions.
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