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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