Chapter 2: Problem 2
Suppose \(\int_{1}^{x} f(t) d t=x^{2}-2 x+1 .\) Find \(f(x)\)
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
\(\sqrt{1}+x\) 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) \(0.5<\int_{0}^{1 / 2} \frac{\mathrm{dx}}{\sqrt{\left(1-\mathrm{x}^{2 \mathrm{a}}\right)}}<0.524\).
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 \(\mathrm{t}^{2}\).
Using Schwartz-Bunyakovsky inequality with \(\mathrm{f}^{2}(\mathrm{x})=\frac{1}{1+\mathrm{x}^{2}}, \mathrm{~g}^{2}(\mathrm{x})=1+\mathrm{x}^{2}\), show that \(\int_{0}^{1} \frac{1}{1+x^{2}} d x>\frac{3}{4}\).
Find the greatest and least values of the function \(\mathrm{I}(\mathrm{x})=\int_{0}^{\mathrm{x}} \frac{2 \mathrm{t}+1}{\mathrm{t}^{2}-2 \mathrm{t}+2} \mathrm{dt}\) on the interval \([-1,1] .\)
Prove that (i) \(\int_{1}^{2} \frac{d x}{x^{3}+3 x+1}<\frac{1}{5}\) (ii) \(3 \sqrt{23}<\int_{2}^{5} \sqrt{3 \mathrm{x}^{3}-1} \mathrm{dx}<10 \sqrt{15}-8 \sqrt{6} / 5\) (iii) \(2<\int_{0}^{4} \frac{d x}{1+\sin ^{2} x}<4\) (iv) \(\frac{\pi}{2}<\int_{0}^{\pi / 2} \frac{\mathrm{d} \theta}{\sqrt{1-\mathrm{k}^{2} \sin ^{2} \theta}}<\frac{\pi}{2 \sqrt{1-\mathrm{k}^{2}}}\left(0<\mathrm{k}^{2}<1\right)\).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
