Question Number 200685 by Spillover last updated on 21/Nov/23 $$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\:\left(\mathrm{1}+\mathrm{tan}{x}\right){dx}\: \\ $$$$ \\ $$ Answered by som(math1967) last updated on 22/Nov/23…
Question Number 200569 by Rupesh123 last updated on 20/Nov/23 Answered by MM42 last updated on 20/Nov/23 $$\bigstar\:\:{e}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}!}\:\Rightarrow\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}!}={e}−\mathrm{2} \\ $$$$\sqrt{{x}\sqrt[{\mathrm{3}}]{{x}\sqrt[{\mathrm{4}}]{{x}\sqrt[{\mathrm{5}}]{\sqrt{{x}…}}}}}={x}^{\frac{\mathrm{1}}{\mathrm{2}}} ×{x}^{\frac{\mathrm{1}}{\mathrm{6}}} ×{x}^{\frac{\mathrm{1}}{\mathrm{24}}}…
Question Number 200570 by Rupesh123 last updated on 20/Nov/23 Answered by witcher3 last updated on 20/Nov/23 $$\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{sin}\left(\mathrm{n}\right)}{\mathrm{n}}=\mathrm{Im}\left\{\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{e}^{\mathrm{in}} }{\mathrm{n}}\right\} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{N}} {\sum}}\mathrm{e}^{\mathrm{in}} =\frac{\mathrm{e}^{\mathrm{i}}…
Question Number 200553 by cortano12 last updated on 20/Nov/23 $$\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\:\sqrt{\mathrm{1}+\mathrm{x}}}\:\mathrm{dx}\:=?\: \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 200604 by Calculusboy last updated on 20/Nov/23 Commented by Frix last updated on 21/Nov/23 $$\mathrm{Does}\:\mathrm{not}\:\mathrm{converge}. \\ $$ Commented by Calculusboy last updated on…
Question Number 200605 by Calculusboy last updated on 20/Nov/23 Commented by Frix last updated on 21/Nov/23 $$\mathrm{Simply}\:\mathrm{integrate}\:\mathrm{by}\:\mathrm{parts}\:\mathrm{to}\:\mathrm{get} \\ $$$$\mathrm{ln}\:\mid\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\mid\:−\frac{{x}}{\mathrm{sin}\:{x}}+{C} \\ $$ Commented by Calculusboy last…
Question Number 200601 by Calculusboy last updated on 20/Nov/23 Commented by Frix last updated on 21/Nov/23 $$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{solution}\:\mathrm{is}\:\mathrm{possible}. \\ $$ Commented by York12 last updated on…
Question Number 200606 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 20/Nov/23 $$\int\frac{\mathrm{cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}}{\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{dx}\:\overset{{t}={x}−\frac{\pi}{\mathrm{4}}} {=} \\ $$$$=\int\left(\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{2}}−\frac{\mathrm{sin}^{\mathrm{2}} \:{t}}{\mathrm{2}}−\frac{\mathrm{tan}\:{t}}{\mathrm{4}}\right){dt}= \\ $$$$=\frac{{t}}{\mathrm{4}}−\frac{\mathrm{cos}^{\mathrm{2}} \:{t}}{\mathrm{4}}−\frac{{t}+\mathrm{cos}\:{t}\:\mathrm{sin}\:{t}}{\mathrm{4}}+\frac{\mathrm{ln}\:\mathrm{cos}\:{t}}{\mathrm{4}}=…
Question Number 200602 by Calculusboy last updated on 20/Nov/23 Answered by Frix last updated on 21/Nov/23 $$\mathrm{Use}\:{u}'=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cos}\:{t}}\:\rightarrow\:{u}=\mathrm{tan}\:\frac{{t}}{\mathrm{2}}; \\ $$$$\:\:\:\:\:\:\:\:\:{v}={t}+\mathrm{sin}\:{t}\:\rightarrow\:{v}'=\mathrm{1}+\mathrm{cos}\:{t} \\ $$$$\int\frac{{t}+\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{cos}\:{t}}{dt}= \\ $$$$=\left({t}+\mathrm{sin}\:{t}\right)\mathrm{tan}\:\frac{{t}}{\mathrm{2}}\:−\int\left(\mathrm{1}+\mathrm{cos}\:{t}\right)\:\mathrm{tan}\:\frac{{t}}{\mathrm{2}}\:{dt}= \\ $$$$={t}\mathrm{tan}\:\frac{{t}}{\mathrm{2}}\:+\mathrm{1}−\mathrm{cos}\:{t}\:−\int\mathrm{sin}\:{t}\:{dt}=…
Question Number 200603 by Calculusboy last updated on 20/Nov/23 Answered by witcher3 last updated on 20/Nov/23 $$\mathrm{x}^{\mathrm{2}} =\left(\mathrm{n}−\mathrm{x}\right)^{\mathrm{2}} −\mathrm{2n}\left(\mathrm{n}−\mathrm{x}\right)+\mathrm{n}^{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{n}−\mathrm{x}\right)^{\mathrm{p}+\mathrm{2}} \mathrm{dx}−\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{n}}…