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Unformatted text preview: Example
Find the antiderivative F (x) of
f (x) = 2
x4 − 9
2x 2 which satisﬁes F (−1) = 10.
Solution: Since 9
f (x) = 2x−4 − x−2 ,
2
it follows from the linearity of antiderivatives and the fact that an antiderivative of xn is
xn+1
given by
when n = −1, that the most general antiderivative F (x) of f (x) is given by
n+1
F ( x) = 2 · x− 3 9 x− 1
2
9
−·
+C =− 3 +
+ C,
−3
2 −1
3x
2x where C is an arbitrary constant. If F must also satisﬁes F (−1) = 10, then
10 = F (−1) = − 2
9
29
+
+ C = − + C,
3
3(−1)
2(−1)
32 and it follows that
C = 10 −
Thus
F (x) = − 29
83
+=.
32
6 2
9
83
+
+.
3x3 2x
6 Note: The answer can always be checked by conﬁrming that
F (x) = d
2
9
83
2
9
− 3+
= 4 − 2 = f (x),
+
dx
3x
2x
6
x
2x and
F (−1) = − 2
9
83
2 9 83
+
+
=−+
= 10.
3
3(−1)
2(−1)
6
32
6 ...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
 Fall '10
 KIHYUNHYUN
 Calculus, Antiderivatives, Derivative

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