Question Number 174421 by ali009 last updated on 31/Jul/22
$$\left.\mathrm{1}\right)\:{show}\:{that} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{−{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}}={ln}\left(\sqrt{\mathrm{5}}−\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right) \\ $$$${determine} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}}} \\ $$$$\left.\mathrm{3}\right)\:{test}\:{the}\:{convergence}\:{of}\:{the}\:{series}\:{given}\:{by} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left({r}+\mathrm{1}\right)!}{{r}!\left({e}^{{r}} \right)} \\ $$$$\left.\mathrm{4}\right) \\ $$$${obtain}\:\mathrm{3}\:{non}\:{zero}\:{terms}\:{of}\:{maclaurrins} \\ $$$${series}\:{for}\:{sin}^{\mathrm{2}} \left({x}\right),{hence}\:{evaluate}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{0}.\mathrm{5}} \frac{{sin}^{\mathrm{2}} \left({x}\right)}{{x}^{\mathrm{2}} }{dx} \\ $$$${given}\:{your}\:{answer}\:{correct}\:{to}\:\mathrm{4}\:{decimal} \\ $$$${places} \\ $$