M138Assign1W06wMaple

# 6 3x 12 find the partial fractions decomposition of

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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) ln + C. 6 (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).] challenge: 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 Send comments or corrections to email:...
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