COMMON equation of integral-differentiation
∫f(x) dx = f(x) + C
OR
f’(x) = f(x) + C
wherein ∫f(x) dx = derivative of f(x)
Basic Definition of integrals?
Integral is the DIFFERENTIATED FORM of a function
INDEFINITE vs DEFINITE INTEGRAL
Indefinite - an f(x) answer and no numbers sa integral
Definite - whole number ang answer and may upper limit and lower limit
8 RULES in indefinite integrals
- constant
- power
- e^x
- ln (x)
- sin (x)
- cos (x)
- constant multiple
- sum and difference rule
ANSWER:
∫ 5dx
∫ 7dy
∫ 8dr
∫ [6z+4] dz
∫ πdn
∫ 7/x^3 dx
- 5x + C
- 7y + C
- 8r + C
- 3 z^2 + 4z + C
- πn + C
- 7 (-1 / 2x^2) + C
MOST important thing in INTEGRALS
+ C
When to use integration by substitution?
If MUKHANG chain rule
INTEGRATION BY SUBSTITUTION STEPS
- Find u
- Find du (derivative of u)
- du = ? dx always
- find value of dx and substitute
EXAMPLE
u = 2x
du = 2 dx
ANSWER:
∫ cos (2x) sin (2x) dx
∫ [ (ln (2x)) / x ]dx
∫ (x^3) / [(2+x^4)^2] dx
u = sin (2x)
u = ln (2x)
u = 2+x^4
Palatandaan sa pag-aassign ng u (3)
- derivative of u can be seen in function and can be cancelled
- more complicated
- higher exponential value
When to use integration by parts? (2)
- if product of 2 functions denoted by the same variable (x e^x)
- integrals of ln or log in simples form (purely lnx or logx)
general equation in INTEGRAL BY PARTS
∫ u dv = u v - ∫ v du
4 main basis in INTEGRAL BY PARTS
u and dv
then
du and v
DEFINITE INTEGRALS
in [a,b]
how do u write the eq?
and how to read this?
(b upper limit) ∫ (a lower limit) f(x) dx = F (b) - F (a)
The (de finite) integral of f(x) with respect to x from a to b
in AREA UNDER PLANE:
Arx
- direction of rectangles
- upper limit and lower limit of integral
- direction of equation : (a-b)
Ary
- direction of rectangles
- upper limit and lower limit of integral
- direction of equation : (a-b)
Arx (y = x variable) find y
- vertical (patayo)
- x-axis so left (lower limit) to right (upper limit) value
- taas - baba
Ary ( x = y variable) find x
- horizontal (pahiga)
- y-axis so baba (lower limit) to taas (upper limit)
- right - left
(x,y,z)
- what do you call this?
- what symbol to denote this?
- how many divisions?
3 dimensional space
R^3
8 - octants
limits for:
- roots
- denominators
- logarithms
- even roots CANNOT contain negative numbers ( bawal yung √-27 or what)
* pero pwede sa odd roots - bawal 0
- bawal 0 and negative numbers
1 possible shape for R^2
3 possible shapes for R^3
R^2 = plane
R^3 = plane, cylinder, surface
Lagrange Extrema step by step
- Write function and constraint
- Write Lagrange EQ ( fx = λgx ; fy = λgy ; constraint)
- Solve for x and y
- Substitute x and y in constraint
- Find value of lambda
- Substitute to x and y eq in #3
- Find max and min values+
