Unformatted text preview: must be a polynomial o f the form 1 ax 2 + b x + C
polynomial
pol
2
For : dy(t) =  g t + u.si n θ
dt
1
y(t) =  g t 2 + u .si n θ . t + may be some constant C.
2
if at time t = 0 the ball was thrown from the ground level ( = 0) : C = 0
if at time t = 0 the ball was thrown from a height of 5 meters: C = + 5m
This mechanical method of calculation wor ks when the functions are
INTEGRABLE. It is known as finding the ANTIDERIVATIVE. There are tables
which you can look up to find the matching ANTIDERIVATIVE of the given
DERIVATIVE. Analyticall
ytically
Analytically, if f(x) is the DERIVATIVE of the function F(x), then the function
F(x) is called the ANTIDERIVATIVE of the function f(x). We may write:
d F(x)
f(x) =
dx
If we write f(x)dx = dF(x) then we call f(x) the DIFFERENTIAL COEFFICIENT.
129 Integ
Below is a partial Table of Integr als of the more frequently encountered functions
deriv tiv
integ
antideriv tiv
integ
in standard form. The derivative = integr and and antiderivative = integr al .
deri
antideri
DERIVA
ANTIDERIVA
F(x)
DERIVATIVE f(x)
ANTIDERIVATIVE F(x)
n+1
x
xn
n ≠ −1
n+1
1
log x
/x
ex
ex
ax
x
a
a > 0, a ≠ 1
log a
sin (x)
− cos x
cos (x)
sin (x)
sec2 (x)
tan (x)
cosec2 (x)
− cot (x)
tan (x) . sec (x)
sec (x)
cot (x) . cosec (x)
− cosec (x)
tan (x)
log sec (x)
cot (x)
log sin (x)
sec (x)
logsec (x) + tan (x)= logtan (π 4 + x/ 2)
/
cosec (x)
logcosec (x) − cot (x)or logtan x/ 2
1
a + x2
1
2
a − x2 1 tan −1 ( x ) a ≠ 0
/a
/a
a+x
1
 log 

2a
a−x 1
x − a2 1
x−a
 log 

2a
x+a 2 2 1 √a2 − x2 x
sin −1(a) x2 < a2
130 26.
26. F(x) = area under f(x) = ∫ f(x)dx
f(x)dx
Let us look at the original concept of the integral  finding the area under a
integral
area
curv
cur ve . For the time being we denote the area function as F(x).
area
curv
F(x) = ar ea under the cur v e of f(x)
In the next chapter we shall prove that :
{f(x)
f(x)}
F(x
ANTIDERIVA
f(x)dx
F( x ) = ANTIDERIVATIVE {f(x)} = ∫ f(x)dx area
We now present 2 methods to find...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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