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Unformatted text preview: he general antiderivative F (x) of a certain rational function f (x) is given by
F (x) = 1
x3 (2x + 5)
(3x − 1)2 Find the partial fractions decomposition of the function f (x) and hence ﬁnd f (x).
[hint: Use log rules ﬁrst to simplify F (x).]
9. Suppose that f is an increasing, diﬀerentiable function for a ≤ x ≤ b, where b > a > 0.
a) Sketch a possible graph of y = f (x) on [a, b]. Outline the rectangular regions with areas
af (a) and bf (b) on the given diagram.
b) Give a geometric argument explaining why
b bf (b) − af (a) = f (b) f −1 (y ) dy. f (x)dx +
a f (a ) (Explain why f −1 exists.)
c) Verify the result in c) as follows:
(i) Use integration by parts to show that, for any diﬀerentiable function f ,
f (x)dx = xf (x) − xf (x) dx. (ii) Integrate the result in (i) from a to b and subsitute x = f −1 (y ) in the right-hand
integral to achieve the ﬁnal result. Math 138 Assignment 1 Addendum
Using Maple for Integration, Partial Fractions and Arc Length
(Classic Version for Windows)
Author: B. A. Forrest
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