Question Number 118436 by bramlexs22 last updated on 17/Oct/20 $$\:\:\int\:\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{2}{x}\right)\:{dx}\: \\ $$ Answered by benjo_mathlover last updated on 17/Oct/20 $$\left(\mathrm{1}\right)\:\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)=\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\left(\mathrm{2}{x}\right)\right)^{\mathrm{2}} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\left(\mathrm{2}{x}\right)+\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\left(\mathrm{4}{x}\right)\right)…
Question Number 118438 by mnjuly1970 last updated on 17/Oct/20 $$\:\:\:\:\:\:…\:\:{nice}\:\:{calculus}… \\ $$$$ \\ $$$$\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${lim}_{{s}\rightarrow\mathrm{0}} \frac{\zeta\left(\:\mathrm{1}+{s}\:\right)+\zeta\left(\mathrm{1}−{s}\right)}{\mathrm{2}}\:\overset{?} {=}\gamma \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\gamma:\:{euler}−{mascheroni}\:{constant} \\ $$$$\:\:\:{m}.{n}.\mathrm{1970}. \\…
Question Number 52900 by MJS last updated on 15/Jan/19 $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mathrm{sin}\:{x}\:\sqrt{\mathrm{sin}\:\mathrm{2}{x}}\:{dx}=? \\ $$$$\underset{−\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:{dx}=? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 15/Jan/19…
Question Number 52898 by MJS last updated on 14/Jan/19 $$\int\mathrm{arcsin}\:{x}\:\mathrm{arccos}\:{x}\:{dx}=? \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 15/Jan/19 $${a}={sin}^{−} {x} \\ $$$${sina}={x} \\ $$$${sin}^{−\mathrm{1}}…
Question Number 52893 by dwdkswd last updated on 14/Jan/19 $$\int\mathrm{sin}\:{x}×\mathrm{cos}\:{x}\:{dx} \\ $$ Answered by MJS last updated on 14/Jan/19 $$\int\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\:{dx}=\frac{\mathrm{1}}{\mathrm{2}}\int\mathrm{sin}\:\mathrm{2}{x}\:{dx}=−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{cos}\:\mathrm{2}{x}\:+{C}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{\mathrm{2}} \:{x}\:+{C}=−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}^{\mathrm{2}} \:{x}\:+{C} \\…
Question Number 118419 by bramlexs22 last updated on 17/Oct/20 $$\:\:\:\int\:\frac{\mathrm{2sin}\:\mathrm{2}{x}}{\mathrm{4cos}\:{x}+\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\: \\ $$ Commented by bramlexs22 last updated on 17/Oct/20 $${yes}…{thank}\:{you}\:{all}\:{master}\: \\ $$ Answered by peter…
Question Number 118395 by TANMAY PANACEA last updated on 17/Oct/20 $$\int_{{a}} ^{{b}} \frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx} \\ $$ Commented by Dwaipayan Shikari last updated on 17/Oct/20 $$\frac{{b}−{a}}{\mathrm{2}} \\…
Question Number 118347 by bramlexs22 last updated on 17/Oct/20 $$\:\:\:\int\:\frac{{dx}}{\:\sqrt{{x}}\:+\sqrt[{\mathrm{3}\:}]{{x}}}\: \\ $$ Answered by benjo_mathlover last updated on 17/Oct/20 $${setting}\:{x}\:=\:\:\:{r}^{\mathrm{6}} \\ $$$$\int\:\frac{\mathrm{6}{r}^{\mathrm{5}} }{{r}^{\mathrm{3}} +{r}^{\mathrm{2}} }\:{dr}\:=\:\:\int\frac{\mathrm{6}{r}^{\mathrm{3}}…
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Question Number 118338 by benjo_mathlover last updated on 17/Oct/20 $$\:\:\:\:{solve}\:\int\:\frac{{dx}}{\mathrm{3}−\mathrm{5sin}\:{x}}\: \\ $$ Answered by Dwaipayan Shikari last updated on 17/Oct/20 $$\int\frac{{dx}}{\mathrm{3}−\mathrm{5}{sinx}} \\ $$$$=\mathrm{2}\int\frac{{dt}}{\mathrm{3}−\frac{\mathrm{10}{t}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}}.\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:{t}={tan}\frac{{x}}{\mathrm{2}}…