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


2) Area: area under f(x) =

Summation Theorems:

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



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





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:


4) Indefinite Integrals:



5) Applications of the Definite Integral:

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




6) Techniques of Integration:


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.


Integrand Substitution Integrand Expression Becomes


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.

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



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

i) ii)

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


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:



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.


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 .


Table of Tests to use on Infinite Series

Test Series Convergence or Divergence Comments
nth Term   Diverges if lim Inconclusive if lim = 0


  (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

(i) Converges if converges

(ii) Diverges if diverges

f (x) must be continuous, positive, decreasing.
Comparison Test

Limit 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 powers

If 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 powers

If 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.


( Calculus 2 MathRoom Index)

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