MATH1051 Lecture Notes - Lecture 26: Product Rule, Ratio Test, Taylor Series
Lecture #26 – Final Review
Integral
●The definition of an integral
●(x)dx Area under the function f(x)"
∫
b
a
f= "
●(x)Δxlim
n→∞ ∑
n
k=1
fk
○In which xΔ = n
b−a
○In which for right endpointsΔxxk=a+k
○In which for left endpointsk)Δxxk=a+ ( − 1
○In which for midpointsk/2)Δxxk=a+ ( − 1
●Remember that for indefinite integrals, don’t forget the +C
●For definite integrals, you evaluate the integral using the bounds as an interval
FTC (Fundamental Theorem of Calculus)
●#1
○(x)dx f(x)(from a to b)
∫
b
a
f′=
○(b) f(a)F−
●#2
○(t)dt g(x)
d
dx ∫
x
a
f=
○Or more specifically, (t)dt g(b(x)b(x)) (a(x)a(x))
d
dx ∫
b(x)
a(x)
g= ′−g′
Properties of Integrals
●(x)dx 0
∫
a
a
f=
●The integral of any function between intervals of same value is
zero
●dx C(b)
∫
b
a
C= − a
●The integral of any constant is , and when intervals arexC
applied, (b)C−a
●(x) (x)dx (x)dx (x)dx
∫
b
a
f±g= ∫
b
a
f± ∫
b
a
g
●Integrals can be added or subtracted in a similar fashion as
derivatives and limits
●f(x)dx C(x)dx
∫
b
a
C= ∫
b
a
f
●Constants can be moved outside of the integral
●(x)dx (x)dx (x)dx
∫
b
a
f= ∫
c
a
f+ ∫
b
c
f
●The integral between a and b is also the integrals between a and c
and c and b added together
●(x)dx (x)dx
∫
b
a
f= − ∫
a
b
f
● Change the sign when flipping the intervals
●or f(x) on [a,b], (x)dx f≥ 0 ∫
b
a
f≥ 0
●or f(x) (x) on [a,b], (x)dx (x)dxf ≥g ∫
b
a
f≥ ∫
b
a
g
●(x) on [a, ], (b)m (x)dx b)Mm ≤f≤M b − a≤∫
b
a
f≤ ( − a
Methods of Integration
●U-substitution
○(g(x))g(x)
∫
f′
○If you see what looks like the result of a chain rule, you use u-sub, which is
considered the un-chain rule method
○ g(x), du g(x)u= = ′
○The result being
○(u)du
∫
f
●Integration by Parts
○dv uv du
∫
u= −∫
v
○If you see what looks like the result of the product rule, use integration by parts,
which is considered the un-product rule method
●Trig-Integrals
