Question Number 133187 by mnjuly1970 last updated on 19/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…….{CALCULUS}…… \\ $$$$\:\:\:\:\:{lim}\:_{{n}\rightarrow\infty} \left({n}\left({ln}\left(\mathrm{2}\right)−\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}+{k}}\right)\right)=? \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 2103 by Yozzi last updated on 02/Nov/15 $${Find}\:{the}\:{integral} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} +\mathrm{1}}. \\ $$ Commented by Yozzi last updated on 02/Nov/15 $${Yup}. \\…
Question Number 133173 by john_santu last updated on 19/Feb/21 $$\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \:\sqrt{\mathrm{1}+\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=? \\ $$ Answered by EDWIN88 last updated on 19/Feb/21 $$\mathrm{I}=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \sqrt{\mathrm{1}+\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}}…
Question Number 67628 by mhmd last updated on 29/Aug/19 $$\int{x}^{{n}} {lnx}/{n}^{{x}\:} \:{dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 2088 by Yozzi last updated on 01/Nov/15 $${Show}\:{that}\:\forall{n}\in\mathbb{Z}^{+} \\ $$$$\int_{\mathrm{0}} ^{\pi/\mathrm{4}} {tan}^{\mathrm{2}{n}+\mathrm{1}} \theta{d}\theta=\left(−\mathrm{1}\right)^{{n}} \left(\frac{\mathrm{1}}{\mathrm{2}}{ln}\mathrm{2}+\underset{{m}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{m}} }{\mathrm{2}{m}}\right). \\ $$$${Deduce}\:{that}\:\underset{\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{m}−\mathrm{1}} }{\mathrm{2}{m}}=\mathrm{0}.\mathrm{5}{ln}\mathrm{2}. \\…
Question Number 2089 by Yozzi last updated on 01/Nov/15 $${If}\:{x}\in\left[\mathrm{0},\mathrm{0}.\mathrm{5}\pi\right],\:{show}\:{that}\:{sinx}\geqslant{x}−\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} . \\ $$$${Hence}\:{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{3000}}\leqslant\int_{\mathrm{0}} ^{\mathrm{1}/\mathrm{10}} \frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}+{sinx}\right)^{\mathrm{2}} }{dx}\leqslant\frac{\mathrm{2}}{\mathrm{5999}}. \\ $$ Commented by prakash jain…
Question Number 67617 by aliesam last updated on 29/Aug/19 Commented by kaivan.ahmadi last updated on 29/Aug/19 $$\int\frac{\sqrt[{\mathrm{4}}]{{x}^{\mathrm{2}} −\mathrm{2}\sqrt[{\mathrm{4}}]{{x}^{\mathrm{6}} }+{x}}}{\:\sqrt[{\mathrm{4}}]{{x}^{\mathrm{3}} }}{dx}=\int\left(\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{{x}}}−\mathrm{2}\sqrt[{\mathrm{4}}]{{x}^{\mathrm{3}} }+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{{x}^{\mathrm{2}} }}\right){dx}= \\ $$$$\int\left({x}^{\frac{−\mathrm{1}}{\mathrm{4}}} −\mathrm{2}{x}^{\frac{\mathrm{3}}{\mathrm{4}}}…
Question Number 67618 by mhmd last updated on 29/Aug/19 $${find}\:{the}\:{area}\:{abovnded}\:{r}={cos}\mathrm{2}\theta \\ $$ Answered by mr W last updated on 29/Aug/19 $${A}=\mathrm{8}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{r}^{\mathrm{2}} {d}\theta}{\mathrm{2}} \\…
Question Number 2069 by Yozzi last updated on 01/Nov/15 $${Find}\:{the}\:{integral} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\left({xcosh}\left({lnx}\right)\right)^{{n}} {dx} \\ $$$${where}\:{n}=\mathrm{0},\mathrm{1},\mathrm{2},… \\ $$ Commented by prakash jain last updated on 01/Nov/15…
Question Number 133125 by abdomsup last updated on 19/Feb/21 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cosx}\:{ch}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$ Commented by mathmax by abdo last updated on 20/Feb/21…