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Chapter 6: Problem 9

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a function. Show that \(f\) is integrable if (i) \(r f\) is integrable for some nonzero \(r \in \mathbb{R}\), or (ii) if \(f\) is bounded, \(f(x) \neq 0\) for all \(x \in[a, b]\), and \(1 / f\) is integrable.

Short Answer

Expert verified
In summary, we have shown that the function \(f(x)\) is integrable if either (i) \(rf(x)\) is integrable for a nonzero constant \(r\) or (ii) \(f(x)\) is bounded, nonzero on \([a, b]\), and \(1/f\) is integrable. For condition (i), we showed that factoring out the constant \(r\) from the integral proves integrability. For condition (ii), we defined function \(g(x) = \frac{1}{f(x)}\), then showed that the integral of \(g(x)f(x)\) exists, guaranteeing the integrability of \(f(x)\).

Step by step solution

01

Prove that if r*f is integrable so is f for nonzero r.

If \(rf\) is integrable, it means that \(\int_{a}^{b}rf(x) \, dx\) exists. Since \(r\) is just a nonzero constant, we can factor it out of the integral. \[\int_{a}^{b}rf(x) \, dx = r \int_{a}^{b}f(x) \, dx\] Now we need to prove that \(f\) is integrable. Since we can factor out the constant \(r\), we can say that the integral of \(f\) exists. Hence, we can conclude that if \(rf(x)\) is integrable, the function \(f(x)\) is also integrable. #Condition (ii)#
02

Show that if f is bounded and nonzero, and 1/f is integrable, then f is integrable.

We are given that \(f\) is bounded, which means there exist constants \(m,M \in \mathbb{R}\) such that: \[m \leq f(x) \leq M, \quad \forall x \in [a,b]\] We are also given that \(f(x) \neq 0\) for all \(x \in [a,b]\), and that \(1/f\) is integrable. It means that \(\int_{a}^{b} \frac{1}{f(x)} \, dx\) exists. To prove that f is integrable, define another function \(g(x) = \frac{1}{f(x)}\) for \(x \in [a,b]\). Now we wish to find if the integral of \(g(x)\) multiplied by f(x) exists. Consider the integral \(\int_{a}^{b} g(x)f(x) \, dx\). Since \(g(x) = \frac{1}{f(x)}\), we have: \[\int_{a}^{b} g(x)f(x) \, dx = \int_{a}^{b} (1) \, dx\]
03

Show f is integrable by showing the integral of g(x)f(x) exists.

Since the integral \(\int_{a}^{b} (1) \, dx\) exists and simply equals \(b-a\), we can conclude that the integrability of \(g(x)f(x)\) is guaranteed. So, if \(1/f\) is integrable, the function \(f(x)\) is also integrable. In conclusion, under two different conditions (i) and (ii), we have proven that the function \(f(x)\) is integrable on the given interval \([a, b]\).

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Most popular questions from this chapter

Let \(a, b, c \in \mathbb{R}\) with \(a

(Taylor's Theorem with Integral Remainder) Let \(n\) be a nonnegative integer and let \(f:[a, b] \rightarrow \mathbb{R}\) be such that \(f^{\prime}, f^{\prime \prime}, \ldots, f^{(n+1)}\) exist and \(f^{(n+1)}\) is continuous on \([a, b]\). Show that $$ f(b)=f(a)+f^{\prime}(a)(b-a)+\cdots+\frac{f^{(n)}(a)}{n !}(b-a)^{n}+\frac{1}{n !} \int_{a}^{b}(b-t)^{n} f^{(n+1)}(t) d t $$ Further, show that the remainder is equal to $$ \frac{(b-a)^{n+1}}{n !} \int_{0}^{1}(1-s)^{n} f^{(n+1)}(a+s(b-a)) d s $$ (Hint: Induction on \(n\) and Integration by Parts.) [Note: The integral remainder does not involve an undetermined number \(c \in(a, b) .]\)

Consider the sequence whose \(n\) th term is given by the following. In each case, determine the limit of the sequence by expressing the \(n\) th term as a Riemann sum for a suitable function. (i) \(\frac{1}{n^{17}} \sum_{i=1}^{n} i^{16}\), (ii) \(\frac{1}{n^{5 / 2}} \sum_{i=1}^{n} i^{3 / 2}\), (iii) \(\sum_{i=1}^{n} \frac{1}{\sqrt{i n+n^{2}}}\), (iv) \(\frac{1}{n}\left\\{\sum_{i=1}^{n}\left(\frac{i}{n}\right)+\sum_{i=n+1}^{2 n}\left(\frac{i}{n}\right)^{3 / 2}+\sum_{i=2 n+1}^{3 n}\left(\frac{i}{n}\right)^{2}\right\\}\).

Let \(m, n \in \mathbb{Z}\) with \(m, n \geq 0 .\) Show that $$ \int_{0}^{1} x^{m}(1-x)^{n} d x=\frac{m ! n !}{(m+n+1) !} $$ (Hint: If \(n \in \mathbb{N}\) and \(I_{m, n}\) denotes the given integral, then using Integration by Parts, \(I_{m, n}=[n /(m+1)] I_{m+1, n-1}\), and \(\left.I_{m+n, 0}=1 /(m+n+1) .\right)\)

(First Mean Value Theorem for Integrals) Let \(f:[a, b] \rightarrow \mathbb{R}\) be a continuous function and \(g:[a, b] \rightarrow \mathbb{R}\) be a nonnegative integrable function. Use the IVP of \(f\) to show that there is \(c \in[a, b]\) such that $$ \int_{a}^{b} f(x) g(x) d x=f(c) \int_{a}^{b} g(x) d x $$ Give examples to show that neither the continuity of \(f\) nor the nonnegativity of \(g\) can be omitted. [Note: For another version of this result, see Exercise 72 .
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