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- INTEGRAL CALCULUS
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- The antiderivative of a function f(x), denoted by F(x), is defined as
-
- ⌠ ,
- │ f(x)dx = F(x) + c where F (x) = f(x) and c is an arbitrary constant.
- ⌡
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- The Fundamental Theorem of Calculus relates the antiderivative to the
- integral as:
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- b
- ⌠ ,
- │ f(x)dx = F(b) - F(a) where F (x) = f(x)
- ⌡
- a
-
- as well as:
- x
- d ⌠
- ── │ f(t)dt = f(x).
- dx ⌡
- a
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- If f(x) ≥ O and continuous on [a,b] then:
- b
- ⌠
- │ f(x)dx is the area under the curve y = f(x) bounded by the
- ⌡
- a
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- lines x = a, x = b and y = O.
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- Properties of integrals of functions f(x) and g(x) continuous
- on [a,b]:
- a
- ⌠
- 1. │ f(x)dx = O
- ⌡
- a
-
- b b
- ⌠ ⌠
- 2. │ k f(x)dx = k │ f(x)dx where k is a constant
- ⌡ ⌡
- a a
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- b b b
- ⌠ ⌠ ⌠
- 3. │ (f(x) ± g(x))dx = │ f(x)dx ± │ g(x)dx
- ⌡ ⌡ ⌡
- a a a
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- b c b
- ⌠ ⌠ ⌠
- 4. │ f(x)dx = │ f(x)dx + │ f(x)dx for c in (a,b).
- ⌡ ⌡ ⌡
- a a c
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- b a
- ⌠ ⌠
- 5. │ f(x)dx = -│ f(x)dx
- ⌡ ⌡
- a b
-
- b
- ⌠
- 6. │ f(x)dx = f(c)(b - a) for some c in [a,b].
- ⌡ (Mean Value Theorem for Integrals)
- a
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- b g(b)
- ⌠ , ⌠ ,
- 7. │ f(g(x)) g (x)dx = │ f(u)du when g (x) is continuous on
- ⌡ ⌡ [a,b]. (u substitution)
- a g(a)
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- Notice that the limits of integration on the u variable correspond to
- those on x through the function u = g(x).
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- The concept of antidifferentiation leads us to the following formula:
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- ⌠ n+1
- │ n x
- 8. │ x dx = ────── + c
- │ n + 1
- ⌡
-
-
- ⌠
- 9. │ cos x dx = sin x + c
- ⌡
-
-
-
- ⌠
- 1O. │ sin x dx = -cos x + c
- ⌡
-
-
- ⌠ 2
- 11. │ sec x dx = tan x + c
- ⌡
-
-
- ⌠ 2
- 12. │ csc x dx = -cot x + c
- ⌡
-
- ⌠
- 13. │ sec x tan x dx = sec x + c
- ⌡
-
-
- ⌠
- 14. │ csc x cot x dx = - csc x + c
- ⌡
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- where c is an arbitrary constant.
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