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Chapter 5 | Integration
If the limits of integration are the same, the integral is just a line and contains no area. 2.
(5.10)
a
f ( x ) dx =− ∫ a b
∫
f ( x ) dx
b
If the limits are reversed, then place a negative sign in front of the integral. 3.
(5.11)
b
b
b
∫
⎦ dx = ∫
f ( x ) dx + ∫
⎡ ⎣ f ( x )+ g ( x ) ⎤
g ( x ) dx
a
a
a
The integral of a sum is the sum of the integrals. 4.
⌠ ⌡ a b
(5.12)
⌡ a b
⎦ dx = ⌠
b
f ( x ) dx − ∫
⎡ ⎣ f ( x )− g ( x ) ⎤
g ( x ) dx
a
The integral of a difference is the difference of the integrals. 5.
(5.13)
b cf ( x ) dx = c ∫ a b
∫
f ( x )
a
for constant c . The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function. 6. (5.14) ∫ a b f ( x ) dx = ∫ a c f ( x ) dx + ∫ c b f ( x ) dx
Although this formula normally applies when c is between a and b , the formula holds for all values of a , b , and c , provided f ( x ) is integrable on the largest interval.
Example 5.11 Using the Properties of the Definite Integral
Use the properties of the definite integral to express the definite integral of f ( x ) =−3 x 3 +2 x +2 over the interval [−2, 1] as the sum of three definite integrals.
Solution Using integral notation, we have ∫ −2 1 ⎛
⎝ −3 x 3 +2 x +2 ⎞
⎠ dx . We apply properties 3. and 5. to get
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