# Properties of the Definite Integral

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Properties of the Definite Integral

a
-1.           f (x)dx = 0
a

a                           b
0.            f (x)dx = −                 f (x)dx
b                           a

b
1.            c dx = c(b − a)
a

b                                       b                             b
2.            f (x) + g(x) dx =                       f (x)dx +                     g(x)dx
a                                       a                             a

b                           b
3.            c f (x)dx = c               f (x)dx
a                           a

b                                       b                             b
4.            f (x) − g(x) dx =                       f (x)dx −                     g(x)dx
a                                       a                             a

c                   b                             b
5.            f (x)dx +           f (x)dx =                     f (x)dx
a                   c                             a

b
6. If f (x) ≥ 0 for all a ≤ x ≤ b, then                                             f (x)dx ≥ 0.
a

b                   b
7. If f (x) ≥ g(x) for all a ≤ x ≤ b, then                                                  f (x)dx ≥           g(x)dx.
a                   a

8. If m ≤ f (x) ≤ M for all a ≤ x ≤ b, then
b
m(b − a) ≤                            f (x)dx ≤ M (b − a).
a

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