Category : Science and Education
Archive   : CAL2.ZIP
Filename : CII_3.HLP

 
Output of file : CII_3.HLP contained in archive : CAL2.ZIP
INTEGRAL CALCULUS



The antiderivative of a function f(x), denoted by F(x), is defined as

ô ,
³ f(x)dx = F(x) + c where F (x) = f(x) and c is an arbitrary constant.
õ

The Fundamental Theorem of Calculus relates the antiderivative to the
integral as:

b
ô ,
³ f(x)dx = F(b) - F(a) where F (x) = f(x)
õ
a

as well as:
x
d ô
ÄÄ ³ f(t)dt = f(x).
dx õ
a


If f(x) ò O and continuous on [a,b] then:
b
ô
³ f(x)dx is the area under the curve y = f(x) bounded by the
õ
a

lines x = a, x = b and y = O.


Properties of integrals of functions f(x) and g(x) continuous
on [a,b]:
a
ô
1. ³ f(x)dx = O
õ
a

b b
ô ô
2. ³ k f(x)dx = k ³ f(x)dx where k is a constant
õ õ
a a

b b b
ô ô ô
3. ³ (f(x) ñ g(x))dx = ³ f(x)dx ñ ³ g(x)dx
õ õ õ
a a a

b c b
ô ô ô
4. ³ f(x)dx = ³ f(x)dx + ³ f(x)dx for c in (a,b).
õ õ õ
a a c

b a
ô ô
5. ³ f(x)dx = -³ f(x)dx
õ õ
a b

b
ô
6. ³ f(x)dx = f(c)(b - a) for some c in [a,b].
õ (Mean Value Theorem for Integrals)
a

b g(b)
ô , ô ,
7. ³ f(g(x)) g (x)dx = ³ f(u)du when g (x) is continuous on
õ õ [a,b]. (u substitution)
a g(a)

Notice that the limits of integration on the u variable correspond to
those on x through the function u = g(x).

The concept of antidifferentiation leads us to the following formula:

ô n+1
³ n x
8. ³ x dx = ÄÄÄÄÄÄ + c
³ n + 1
õ


ô
9. ³ cos x dx = sin x + c
õ



ô
1O. ³ sin x dx = -cos x + c
õ


ô 2
11. ³ sec x dx = tan x + c
õ


ô 2
12. ³ csc x dx = -cot x + c
õ

ô
13. ³ sec x tan x dx = sec x + c
õ


ô
14. ³ csc x cot x dx = - csc x + c
õ

where c is an arbitrary constant.


  3 Responses to “Category : Science and Education
Archive   : CAL2.ZIP
Filename : CII_3.HLP

  1. Very nice! Thank you for this wonderful archive. I wonder why I found it only now. Long live the BBS file archives!

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