Example
Find the antiderivative
F
(
x
) of
f
(
x
) =
2
x
4

9
2
x
2
which satisfies
F
(

1) = 10.
Solution:
Since
f
(
x
) = 2
x

4

9
2
x

2
,
it follows from the linearity of antiderivatives and the fact that an antiderivative of
x
n
is
given by
x
n
+1
n
+ 1
when
n
6
=

1, that the most general antiderivative
F
(
x
) of
f
(
x
) is given by
F
(
x
) = 2
·
x

3

3

9
2
·
x

1

1
+
C
=

2
3
x
3
+
9
2
x
+
C,
where
C
is an arbitrary constant. If
F
must also satisfies
F
(

1) = 10, then
10 =
F
(
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 Fall '10
 KIHYUNHYUN
 Calculus, Antiderivatives, Derivative, Fundamental Theorem Of Calculus, +C =−

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