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Unformatted text preview: a, b .
Since the function log is uniformly continuous on the interval 1, Ý , the integrability of g follows at
once from the composition theorem.
4. Suppose that f is integrable with respect to an increasing function on an interval a, b and that f x
for every x
a, b . Show how the junior version of the composition theorem for integrability can be used to
show that if h is any continuous function on the interval , then the function h f is integrable with
respect to on a, b .
The result follows at once from the fact that any continuous function on the interval , must be
uniformly continuous. Exercises on the Change of Variable Theorem
1. a. Given that f is a continuous function on the interval
2 Þ0 Solution: We define u x 1, 1 , prove that f sin x cos xdx 0. sin x for every number x
2 Þ0 f sin x cos xdx
2 Þ0 0, 2 and observe that f u x u x dx u 2 Þu 0 f Þ0 f 0
0 b. Given that f is a continuous function on the interval 0, 1 , prove that
/2 Þ0 Solution: We define u x
/2 Þ0 f sin x dx Þ /2 f sin x dx. x for every number x
/2 f sin x dx Þ0 Þ0 Þu 0 /2 f sin
2 and observe that x u x dx f sin u x u /2 0, u x dx f sin t dt /2 Þ f sin t dt Þ /2 f sin t dt Of course, it makes no difference whether we write t or x in the integral Þ
c. Given that 0, prove that
/2 Þ0 Solution: We define u x sin xdx 2 Þ 2x for all x 314 /2 sin x cos xdx. 0 0, /2 and observe that
/2 f sin t dt. 2 Þ /2
0 sin x cos xdx /2 Þ0 /2 Þ0 2 sin x cos xdx
/2 1
2 Þ0 1
2 Þu 0 /2 Þ0 sin 2x 2dx 1
2 u /2 2 sin x cos x dx
sin u x u x dx Þ 0 sin tdt sin tdt 1
2 /2
1 Þ sin tdt Þ sin tdt
2
0
/2
and from part b we deduce that the latter expression is equal to 1
2 /2 Þ0 sin tdt Þ /2 sin tdt /2 Þ0 0 sin tdt. 2. Given that u is a differentiable function on an interval a, b and that its derivative u is integrable on a, b
and given that u a u b and that f is integrable on the range of u, prove that Þa f u t
b u t dt 0. From the change of variable theorem we see that Þa f u t
b Þu a ub u t dt Þu a ua f x dx f x dx 0. 3. Given that f is integrable on an interval a, b and that c is any number, prove that
b c Þ a f t dt Þ ac f t
b c. We observe that u t 1 for every t. Now For every number t we define u t t
b c Þ a c f t c dt c dt b c Þ a c f u t u t dt Þu a ub f x dx Þ a f t dt.
b I have changed the name of the dummy variable back to t to match the expression in the exercise.
4. Given that a, b and c are real numbers, that ac bc and that f is a continuous function on the interval
ac, bc , prove that Þ ac f t dt c Þ a f ct dt.
bc b Hint: Look at the definition u t ct for each t.
We assume that c 0. We define u t ct for every number t. We see that u t c for each t and
we deduce from the change of variable theorem that
c Þ f ct dt
b a 5. Þa f u t
b u t dt Þ ac f x dx Þ ac f t dt...
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 Fall '08
 STAFF
 Math, Calculus

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