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Chapter 10: Problem 8
Let \(f(x):=\cos x\) for \(x \in[0, \infty)\). Show that \(f \notin \mathcal{R}^{*}[0, \infty)\).
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If \(f, g \in \mathcal{L}[a, \infty)\), show that \(f+g \in \mathcal{L}[a, \infty)\). Moreover, if \(\left.\|h\|:=\int_{a}^{\infty} \mid h\right]\) for any \(h \in\) \(\mathcal{L}[a, \infty)\), show that \(\|f+g\| \leq\|f\|+\|g\|\).
Use the Substitution Theorem \(10.1 .12\) to cvaluate the following integrals. (a) \(\int_{-3}^{3}(2 t+1) \operatorname{sgn}\left(t^{2}+t-2\right) d t=6\), (b) \(\int_{0}^{4} \frac{\sqrt{t} d t}{1+\sqrt{t}}\). (c) \(\int_{1}^{5} \frac{d t}{t \sqrt{t-1}}=2 \operatorname{Arctan} 2\), (d) \(\int_{0}^{1} \sqrt{1-t^{2}} d t\).
Establish the convergence or the divergence of the following integrals: (a) \(\int_{0}^{\infty} \frac{\ln x d x}{x^{2}+1}\). (b) \(\int_{0}^{\infty} \frac{\ln x d x}{\sqrt{x^{2}+1}}\), (c) \(\int_{0}^{\infty} \frac{d x}{x(x+1)}\), (d) \(\int_{0}^{\infty} \frac{x d x}{(x+1)^{3}}\), (e) \(\int_{0}^{\infty} \frac{d x}{\sqrt[3]{1+x^{3}}}\), (f) \(\int_{0}^{\infty} \frac{\operatorname{Arctan} x d x}{x^{3 / 2}+1}\).
Let \(\delta\) be a gauge on \([a, b]\) and let
\(\dot{\mathcal{P}}=\left\\{\left(\left[x_{i-1}, x_{i}\right],
t_{i}\right)\right\\}_{i=1}^{n}\) be a \(\delta\) -fine partition of \([a, b]\).
(a) Show that \(0
Show that if \(n \in \mathbb{N}, s>0\), then \(\int_{0}^{\infty} x^{n} e^{-s x} d x=n ! / s^{n+1}\).
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