Calculus 2 Tool Box

1) Antidifferentiation:

a) Power Rule: If f (x) = ax n then

b) Trig Functions:

 derivative f(x) antiderivative F(x) cos u du sin u + C sin u du – cos u + C sec 2 u du tan u + C csc u cot u du – csc u + C sec u tan u du sec u + C csc 2 u du – cot u + C c) Exponentials and Logs: derivative f (x) antiderivative F(x) e u du e u + C a u du ln u + C

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2) Area: area under f(x) =

Summation Theorems:

 a) b) c) d) e) f) g)

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3) The Definite Integral:

a) Definition:

b) Theorems on the Definite Integral:

i) If a > b, then

ii) = the area under f (x) if f is integrable (continuous) and

iii)

iv)

v)

vi)

c) The Mean Value Theorem for Integrals:

If f (x) is continuous on [a, b], then there exists a z in (a, b) such that

That is: area of rectangle with height = f (z), base = (b – a) = integral value or area under f (x).

** NOTE: z cannot be an endpoint of the interval [a, b]. It must belong to the open interval.

d) The Fundamental Theorem of Calculus:

If f (x) is continuous on [a, b], then:
1) is a general antiderivative of f on [a, b]
2) = F(b) – F(a) where F is the antiderivative of f .

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4) Indefinite Integrals:

a)
b) method of substitution:
when we substitute u = g(x) and du = g /(x) dx

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5) Applications of the Definite Integral:

a) Area: If , and both f (x) and g (x) are continuous on [a, b], then
= the area between the curves f (x) and g (x) on [a, b].

b) Volumes of Solids of Revolution:
i) Disk Method:
If we revolve about the x-axis
If we revolve about the y-axis
Both these equations are the equivalent of , the formula for the volume of a cylinder, since each disk we cut is a cylinder with radius = f (x) and height = dx.
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If there is a hole in the center of the disk, f is the outer radius and g is the inner radius, so:
If we revolve about the x-axis
If we revolve about the y-axis
ii) Shell Method:
If we revolve about the x-axis
If we revolve about the y-axis

**NOTE: Always use the variable corresponding to the axis on which the shells are standing.

If the bottom of the shells are defined by g(x) then:
If we revolve about the x-axis
If we revolve about the y-axis

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c) Volumes by Slicing:
if the slices are perpendicular to the x-axis
if the slices are perpendicular to the y-axis
d) Arc Length:
 or
depending on whether you use f (x) or f (y).

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6) Techniques of Integration:

a) by parts:
this is just the integration equivalent of the product rule.
b) Trig Integrals:

odd powers: reduce n by 1 and reserve for du,

change the even powers using sin 2 u + cos 2 u = 1

even powers: use ½ angle formulas.

2) :

n is even: reduce n by 2, use sec 2 x dx as du, change sec n – 2 x using sec 2 x = 1 + tan 2 x

m is odd : use sec x tan x dx as du, change the rest using tan 2 x = sec 2 x – 1.

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c) Trig Substitutions:

 Integrand Substitution Integrand Expression Becomes

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d) Partial Fraction Decomposition:

i) If the denominator factor is (px + q)k , make fractions of the form for n = 1, 2 .... k.

ii) If the denominator factor is an irreducible quadratic of the form (ax2 + bx + c)k,
make fractions of the form for n = 1, 2, 3 .......... k.

e) Quadratic Expressions:

If the integrand has an irreducible quadratic as denominator,
complete the square and use substitution.

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f) Miscellaneous Substitutions:
i) for integrands with the nth root of f (x), use either u = [f (x)]1/ n or u n = f (x).
ii) for integrands with x1/ n and x1/ m , substitute u m n = x.

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7) Indeterminate Forms, L'Hopital's Rule, Improper Integrals:

a) If , use l'Hopital's rule ie:
b) If using l'Hopital's rule.
c) If becomes , take ln y and use the approach described
in (a) or (b).
Don't forget to evaluate y and not ln y at the end of the question.
d) when you have , and the terms are fractions,
find the common denominator, combine the 2 fractions, use the approach of part (a).
e) Integrals with Infinite Limits of Integration:

 i) ii) iii)

** f (x) must be continuous over the interval of integration. Set a = 0 for efficiency if possible.

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f) Integrals with Discontinuous Integrands:

i) If f (x) is discontinuous at b , then

ii) If f (x) is discontinuous at a , then

iii) If f (x) is discontinuous at c where a < c < b, then:

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8) Infinite Series:

Definition:

There are 3 kinds of infinite series:

 1) positive term series, 2) alternating series, 3) varied sign series.

Following is a table with a list of many, many series tests to use when testing for convergence.

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Absolute Convergence:

An infinite series is absolutely convergent if the series of absolute values is convergent. Generally, we use the Ratio Test to show absolute convergence.

Example: the series includes both positive and negative terms but it is not alternating. Since we know that , and we know that n² is always > 0, we can compare it to , a convergent p-series with p = 2. By the basic comparison test, this series is absolutely convergent.

Conditional Convergence :

Some series that include both positive and negative signs can be shown to be convergent in one form but divergent in another. The classic example is the harmonic series which is divergent as a positive term series but convergent as an alternating series .

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Table of Tests to use on Infinite Series

 Test Series Convergence or Divergence Comments nth Term Diverges if lim Inconclusive if lim = 0 GeometricSeries (i) Converges if | r | < 1(ii) Diverges if | r | 1 If convergent, the sum is Useful for comparison tests p-series (i) Converges if p > 1(ii) Diverges if p 1 Useful for comparison tests Integral (i) Converges if converges(ii) Diverges if diverges f (x) must be continuous, positive, decreasing. Comparison TestLimit Comparison test test , select both positive (i) converges if converges and for all values of n.(ii) diverges if diverges and for all values of n.(iii) If lim , then both series converge or both series diverge. The comparison series is often a geometric or a p-series. Ratio Test If lim ,:(i) converges if L < 1(ii) diverges if L > 1 Inconclusive if L = 1.Useful if an involves factorials or nth powersIf all a are > 0, ignore the absolute value signs. Root Test If , the series:(i) converges if L < 1(ii) diverges if L > 1 Inconclusive if L = 1.Useful if an involves only nth powersIf all a are > 0, ignore the absolute value signs. Alternating Series Test an > 0 Converges if for every k and Applies only to an alternating series. Absolute Convergence Test If is convergent then is convergent Useful for series that contain both positive & negative terms.

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