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Chapter 2: Problem 1
Find the sum of the series \(\frac{x^{2}}{1.2}-\frac{x^{3}}{2.3}+\frac{x^{4}}{3.4}-\ldots+(-1)^{n+1} \frac{x^{n+1}}{n(n+1)}+\ldots,|x|<1\)
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Let \(\mathrm{F}(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}\). Determine a formula for computing \(\mathrm{F}(\mathrm{x})\) for all real \(\mathrm{x}\) if \(\mathrm{f}\) is defined as follows: (a) \(\mathrm{f}(\mathrm{t})=(\mathrm{t}+\mid \mathrm{t})^{2}\) (b) \(f(t)=\left\\{\begin{array}{lll}1-t^{2} & \text { if } & |t| \leq 1 \\\ 1-|t| & \text { if } & |t|>1\end{array}\right.\) (c) \(\mathrm{f}(\mathrm{t})=\mathrm{e}^{-1}\). (d) \(\mathrm{f}(\mathrm{t})=\) the maximum of 1 and \(\mathrm{t}^{2}\).
\(\sqrt{1}+x\) Prove that, if \(\mathrm{n}>1\) (i) \(0<\int_{0}^{\pi / 2} \sin ^{n+1} x d x<\int_{0}^{\pi / 2} \sin ^{n} x d x\), (ii) \(0<\int_{0}^{\pi / 4} \tan ^{n+1} x d x<\int_{0}^{\pi / 4} \tan ^{n} x d x\). (iii) \(0.5<\int_{0}^{1 / 2} \frac{\mathrm{dx}}{\sqrt{\left(1-\mathrm{x}^{2 \mathrm{a}}\right)}}<0.524\).
\begin{aligned} &\text { Integrating by parts, prove that }\\\ &0<\int_{100 \pi}^{200 \pi} \frac{\cos \mathrm{x}}{\mathrm{x}} \mathrm{dx}<\frac{1}{100 \pi} \end{aligned}
Let \(\mathrm{f}\) be twice continuously differentiable in \([0,2 \pi]\) and concave up. Prove that \(\int_{0}^{2 \pi} f(x) \cos x d x \geq 0\)
Show that \(\int_{0}^{1} \frac{\ell n\left(1-a^{2} x^{2}\right)}{x^{2} \sqrt{\left(1-x^{2}\right)}} d x\) \(=\pi\left[\sqrt{1-a^{2}}-1\right],\left(a^{2}<1\right)\)
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