calcolous 2

Chapter 9 Techniques of integration

9.1 Integration by Parts£ udv= uv -£ v du

9.2 Trigonometric Integrals guidelines for Evaluationsin ^m cos^n dx

9.3 guidelines for Evaluation£tan ^m X sec^n X dx

9.3 Trigonometric Substitution

9.4 Integrals of Rational Function example one and sample 4

9.5Integrals Involving Quadratic provesample 1 page 479

Miscellaneous Substitution(f(x))1/n substitution u=(f(x))1/n or u = f(x)

9.7 Tables of IntegralMathematicians and scientist who use Integrals sometimes use tables to solve them. appendix IV page A21-A26

10. Forms and Improper Integrals.

10.1 The Indeterminate forms 0/0 & infinity/infinity.g

Couchy's formula and proof

LHopital Rule

10.2 Other Indeterminate formsguidelines for Investigating lim x➡ c (f (x) g (x)) for the form 0 to infinity

Investigating limiters involving exponential expressions. Table.

10.4 guidelines for Investigating lim x➡ c (f (x)^g (x)) for the forms 0^0, 1^ infinity to infinity^0

last indeterminate forms.

Subtopic

10.3 Integral with infinite limits of integration f is continues and nonnegative on an infinity interval ( a, to inf) and lim x-infinity f (x)= 0 if t>a, en the area A(t) under the graph of f from a to t, A(t) = £f(x) dx

definition 10.6

Subtopic

10.4 Integrals with discontinuous IntegrandsIf a function F is continuous on a close interval, then then definite integral exists. if F an infinity discontinuity at some number in the interval, it may still be possible to assign a value to the integral.

11. infinity series

11.1Definition 11.2 and graph 11.1

Definition11.3Figure 11.2

Definition 11.4 and theorem 11.5

Subtopic

Theorem 11.6 proof to start with Definition 11.3

Sandwich theorem for sequences

Theorem 11.8

monotonic: nondecreasing or non increasing term series.Theorem 11.9: A bounded, monotonic sequence has a limit. Completeness property: If a non empty set S of real numbers has a upper bound, then S has a least upper bound. proof 11.9 and graph 11.7

11.2 Convergent or Divergent Series

Definition 11.4 theorem 11.5 include proof 

theorem 11.16 and proof & n-th- term test

Subtopic

Theorem 11.18 and proof theorem 11.19 and proof

Theorem 11.20

Theorem 11.21 and proof 

11.3 positive- term seriesand proof

Integral test and proofsandwich theorem

P- series or hyper-harmonic series theorem 11.25; use the integral test

basic comparative test and proof

limit comparison test and proof

Theorem 11.28 and proof 

Root test

Chapter 7
Logarithmic and Exponential Functions

7.1
Inverse Functions
a function F with a domain D and range R is a one-to-one function if whenever a is not equal to b in D, then f(a) is not equal to f (b) in R. then f has an inverse function call g if only if g has a domain R and range D.

I. g(f(x))= to x for everyone in D.
II. f(g(y)= y for every y in R.

D of F-1 = range of f
R of F-1= domain of f

finding f-1
-Verify f is a one-to-one function ( increase or decrease domain.)
-Solve the equation; in terms of y and x= f-1 (y)
-Verify the two condition
f-1 f(x) = x and f(f-1(x)) = x
sample 1 (p376)

theorem 7.7
g'(c) = 1/f'g(c)

Corollary 7.8
g is the -1 function of a Dx to F and if f'(g(x) not equal to 0
then g'(x) = 1 / f'(g(x)

7.2 The Natural Logarithmic Function (ln)

Theorem
u= g(x) and g is Dx then.
I. Dx ln u =1/u Dx u if g(x)>0
II.Dx ln |u| =1/u Dx u if g(x) not equal to 0

Laws of Natural Logarithms.
if p>0 and q > 0, then
I. ln pq = ln p + ln q
II. ln p\q = ln p - ln q
III. ln p^r = r ln p for evere rational number r
( in this exercise is better to convert the function to the LNL and work the rationals as exponents before differentiating.

Guidelines for Logarithmic differentiation (5 ways )
page 388.

7.3 The Natural Exponential function

Theorem 7.20
I. e^p* e^q=e ^p+q.
II. e^p/e^q = e^p-q.
III. (e^p)^ = e^pr.

denoted by " exp," is the inverse of the natural logarithmic function. (e= 2.71828)
ln e^r = r ln e = r(1)= r
I. ln e^x = x for every x
II.e^lnx = x for every x >0

Theorem 7.21
Dx e^ =e^x

Theorem 7.22
If u = g(x) is Dx then
Dx e^u = e^u Dx u.
example 2 and 3 (p395-396)

7.4 Integration

If u = g(x) is not equal to 0 and g is Dx, then
£1/u du = ln |u| + c

definite integral use an indefinite integral to find an antiderivative. Use u to substitute and du. work the equation to make it work in u and du and then apply the fundamental theorem of calculus.

Theorem 7.24
if u =g(x) and g is Dx, then
£ e^u du = e^u + c
sample 5 & 6. (p 403)

Theorem 7.25 integration of trigonometry functions that are logarithmic functions. page (404)samples 7 & 8 page (405)-In order to make to solve the integral we need to use substitution (£ cos u du) - £ tan u du to replace and give value (top - bottom) - when substitute by u. Find values of u, when using the values of x.

7.5 General Exponential And Logarithmic Functions

definition 7.26 a* = e^xlna for every a >0 and every real number x.LAWS OF EXPONENTS let a>0 and b > 0 if u and v are many real numbers, then a^u a^v = a^u+v (a^u)^v = a^uv(ab)^u a^u b^va^u/a^v=a^u-v(a/b)^u = a^u/b^u

theorem 7.28 (i)Dx a^x = a^x ln a (ii) Dx a^u= a^u ln a Dx u

theorem 7.29 (i) £,a^dx = (1/lna) a^x +C (ii) £ a^u du (1/lna ) a^u + C

Definition of a log x y= log x if and only if x= a ^ ytheorem 7.31 (i) Dx loga x= Dx (lnx/lna) = 1/lna 1/x (ii) Dx loga |u|= Dx (ln |u|/lna) = 1/lna 1/u Dx u theorem (i) lim (1+ h) ^ 1/h = e when h~ 0(ii) lim (1+ 1/n) ^ n= e when n~ infinity.

Laws of Growth and Decaydy/dt = Cy or dy = cy dttheorem 7.33let y be differentiable function of t such that y> 0 for every t, and let y0 be the value y at t= 0 if dy/dt = cy for some constant c  y= y0e^ct

Chapter 8 Inverse Trigonometry and Hyperbolic Functions

8.1 the inverse sin function (sin^-1) is defined by y= sin ^-1x if and only if x= sin y (for every -1<~x<~1 and -p/2<~y<~p/2properties of sin-1 page 427the inverse sin function (cos^-1) is defined by y= cos^-1x if and only if x= cos y (for every -1<~x<~1 and 0<~y<~pproperties of cos^-1 page 428the inverse sin function (tan^-1) is defined by y= tan ^-1x if and only if x= tan y (for every x for every x and. -p/2<y<p/2properties of tan^-1 page 429.