Integrals#

Indefinite integrals#

Description

Equations

Indefinite integral (antiderivative)

F(x)=f(x) dxF(x)=f(x)

Antiderivative as a family of functions
(Plus C!)

If F is an antiderivative of f, C is a constant,
then the most general antiderivative is F(x)+C

Table of indefinite integrals#

Function f(x)

Antiderivative F(x)

Function f(x)

Antiderivative F(x)

xn

xn+1n+1+C

1x

ln|x|+C

ex

ex+C

bx

bxlnb+C

sinx

cosx+C

cosx

sinx+C

1x2+a2

1aarctan(xa)+C

1a2x2

arcsin(xa)+C

Definite integrals as Riemann sums#

Description

Equations

Area

A=limnRn=limni=1nf(xi)Δx

Definite integral

abf(x) dx=limni=1nf(xi)Δx

Operational definition of definite integral as Riemann sum

abf(x) dx=limni=1nf(xi)Δx
Δx=banxi=a+iΔx

Properties of definite integrals#

Description

Equations

Reversing the bounds changes the sign of definite integrals

abf(x) dx=baf(x) dx

Definite integral is zero if upper and lower bounds are the same

aaf(x) dx=0

Definite integrals of constant

abc dx=c(ba)

Addition and subtraction of definite integrals

ab[f(x)±g(x)] dx=abf(x) dx±abg(x) dx

Constant multiple of definite integrals

abcf(x) dx=cabf(x) dx

Fundamental theorem of calculus#

Description

Equations

Fundamental Theorem of Calculus I
(If f is continuous on [a,b])

g(x)=axf(t)dt
g(x)=f(x)

Fundamental Theorem of Calculus II
(If f is continuous on [a,b])

abf(x)dx=F(b)F(a)
where F is any antiderivative of f

Net Change Theorem
(The integral of a rate of change is the net change)

abF(x)dx=F(b)F(a)

Substitution rule#

Description

Equations

Substitution rule (u-substitution)
ug(x)

f(g(x))g(x) dx=f(u) du

Substitution rule for definite integrals
ug(x)

abf(g(x))g(x) dx=g(a)g(b)f(u) du

Integrals of even functions

aaf(x) dx=20af(x) dx

Integrals of odd functions

aaf(x) dx=0

Techniques of Integration#

Integration by parts#

Description

Equations

Integration by parts

f(x)g(x) dx=f(x)g(x)g(x)f(x) dx

Integration by parts

u dv=uvv du

Integration by parts for definite integrals

abfg dx=[fg]ababfg dx

Trigonometric integrals#

Description

Equations

Integral of even power of sine and cosine using trig identities

sin2(x)=12(1cos(2x))cos2(x)=12(1+cos(2x))sin(x)cos(x)=12sin(2x)

Trig identity for solving
sin(mx)cos(nx)dx

sinAcosB=12[sin(AB)+sin(A+B)]

Trig identity for solving
sin(mx)sin(nx)dx

sinAsinB=12[cos(AB)cos(A+B)]

Trig identity for solving
cos(mx)cos(nx)dx

cosAcosB=12[cos(AB)+cos(A+B)]