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Category: Integration

nice-calculus-i-1-4-9-9-36-16-100-ii-0-pi-2-x-2-cot-x-dx-m-n-1970-

Question Number 117806 by mnjuly1970 last updated on 13/Oct/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:\:\:{i}\:::\:\:\:\mathrm{1}\:+\frac{\mathrm{4}}{\mathrm{9}}+\frac{\mathrm{9}}{\mathrm{36}}+\frac{\mathrm{16}}{\mathrm{100}}+…=\:?? \\ $$$$\:\:\:\:\:{ii}::\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}^{\mathrm{2}} {cot}\left({x}\right)\:{dx}=?? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$$$ \\ $$ Commented by…

pi-2-0-dx-cox-x-2-cos-x-2-2-cos-x-2-n-

Question Number 183320 by mathlove last updated on 25/Dec/22 $$\underset{\mathrm{0}} {\int}^{\frac{\pi}{\mathrm{2}}} \frac{{dx}}{{cox}\frac{{x}}{\mathrm{2}}\centerdot{cos}\frac{{x}}{\mathrm{2}^{\mathrm{2}} }\centerdot\centerdot\centerdot\centerdot\centerdot{cos}\frac{{x}}{\mathrm{2}^{{n}} }}=? \\ $$ Answered by Vynho last updated on 28/Dec/22 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…

sin-x-cos-x-tan-x-cot-x-3-dx-

Question Number 183279 by cortano1 last updated on 24/Dec/22 $$\:\:\int\:\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\left(\mathrm{tan}\:{x}−\mathrm{cot}\:{x}\right)^{\mathrm{3}} }\:{dx}\:=? \\ $$ Answered by MJS_new last updated on 24/Dec/22 $$\mathrm{with}\:{t}=\mathrm{tan}\:\frac{{x}}{\mathrm{2}}\:\mathrm{we}\:\mathrm{get} \\ $$$$−\mathrm{16}\int\frac{{t}^{\mathrm{3}} \left({t}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}}…

sin-1-x-x-2-dx-

Question Number 117724 by bemath last updated on 13/Oct/20 $$\int\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$ Commented by MJS_new last updated on 13/Oct/20 $$\mathrm{you}\:\mathrm{lost}\:\mathrm{the}\:“−''\:\mathrm{in}\:\mathrm{line}\:\mathrm{5} \\ $$$$\mathrm{btw}.:\:\mathrm{tan}\:\frac{\mathrm{arcsin}\:{x}}{\mathrm{2}}\:=\frac{{x}}{\mathrm{1}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}…

sin-x-cos-x-sin-x-cos-x-dx-

Question Number 183243 by cortano1 last updated on 24/Dec/22 $$\:\:\int\:\frac{\sqrt{\mathrm{sin}\:{x}}\:−\sqrt{\mathrm{cos}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}\:+\:\sqrt{\mathrm{cos}\:{x}}}\:{dx}\:=? \\ $$ Answered by MJS_new last updated on 24/Dec/22 $$\int\frac{\sqrt{\mathrm{sin}\:{x}}−\sqrt{\mathrm{cos}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}+\sqrt{\mathrm{cos}\:{x}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{tan}\:{x}}\:\rightarrow\:{dx}=\frac{\mathrm{2}{t}}{{t}^{\mathrm{4}} +\mathrm{1}}{dt}\right] \\ $$$$=\mathrm{2}\int\frac{{t}\left({t}−\mathrm{1}\right)}{\left({t}+\mathrm{1}\right)\left({t}^{\mathrm{2}}…

Question-117654

Question Number 117654 by jeewanbee640 last updated on 13/Oct/20 Commented by prakash jain last updated on 13/Oct/20 $$\left({ii}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{have}\:\mathrm{period}\:\mathrm{2}\pi \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$$${f}\left(\mathrm{2}\pi+\mathrm{1}\right)=\mathrm{8}\pi+\mathrm{4}\neq{f}\left(\mathrm{1}\right) \\ $$ Terms…