Question Number 85718 by M±th+et£s last updated on 24/Mar/20 $$\int\frac{{sin}\left({x}\right)−{cos}\left(\mathrm{3}{x}\right)}{{sin}\left({x}\right)−{cos}\left(\mathrm{2}{x}\right)}{dx} \\ $$ Answered by MJS last updated on 24/Mar/20 $$\int\frac{\mathrm{sin}\:{x}\:−\mathrm{cos}\:\mathrm{3}{x}}{\mathrm{sin}\:{x}\:−\mathrm{cos}\:\mathrm{2}{x}}{dx}= \\ $$$$=\int\frac{\mathrm{4cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:−\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{−\mathrm{2cos}^{\mathrm{2}} \:{x}\:+\mathrm{sin}\:{x}\:+\mathrm{1}}{dx}= \\…
Question Number 85717 by M±th+et£s last updated on 24/Mar/20 $$\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\mathrm{4}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:{dx} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{10}} \left(\mathrm{1}−{x}^{{n}} \right){dx} \\ $$$$ \\…
Question Number 85711 by Rio Michael last updated on 24/Mar/20 $$\:\underset{−\mathrm{4}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{2}{x}\:+\:\mathrm{1}}{\left({x}^{\mathrm{2}} +\:{x}\:+\:\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }\:{dx} \\ $$ Commented by jagoll last updated on 24/Mar/20 $$\underset{−\mathrm{4}}…
Question Number 151246 by mathmax by abdo last updated on 19/Aug/21 $$\mathrm{find}\:\mathrm{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{cosx}\right)\mathrm{dx}\:\mathrm{and}\:\mathrm{J}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{sinx}\right)\mathrm{dx} \\ $$ Answered by qaz last updated on 19/Aug/21…
Question Number 20166 by tammi last updated on 23/Aug/17 $$\int\mathrm{cosec}\:^{\mathrm{2}} {xdx} \\ $$ Answered by sma3l2996 last updated on 23/Aug/17 $${t}={cosecx}\Rightarrow{dt}=−\frac{{cosec}^{\mathrm{2}} {x}}{{secx}}{dx} \\ $$$${cosec}^{\mathrm{2}} {xdx}=−\frac{{tdt}}{\:\sqrt{{t}^{\mathrm{2}}…
Question Number 20167 by tammi last updated on 23/Aug/17 $${please}\:{solve}\:{it} \\ $$$${integrate}\:{with}\:{respect}\:{to}\:{x} \\ $$$$\int\frac{\mathrm{5}{x}−\mathrm{2}}{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}} \\ $$ Answered by ajfour last updated on 23/Aug/17 $$=\frac{\mathrm{1}}{\mathrm{3}}\int\frac{\left(\mathrm{5}/\mathrm{2}\right)\left(\mathrm{2}{x}+\mathrm{2}/\mathrm{3}\right)−\mathrm{11}/\mathrm{3}}{\left({x}+\mathrm{1}/\mathrm{3}\right)^{\mathrm{2}}…
Question Number 20164 by tammi last updated on 23/Aug/17 $$\int\frac{{e}^{\mathrm{tan}^{−\mathrm{1}} {x}} }{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$ Answered by ajfour last updated on 23/Aug/17 $${let}\:\mathrm{tan}^{−\mathrm{1}} {x}={t} \\…
Question Number 85701 by jagoll last updated on 24/Mar/20 $$\int\:\frac{\sqrt{\mathrm{3x}−\mathrm{1}}}{\:\sqrt{\mathrm{2x}+\mathrm{1}}}\:\mathrm{dx}\: \\ $$ Commented by john santu last updated on 24/Mar/20 $${let}\:{t}\:=\:\sqrt{\mathrm{2}{x}+\mathrm{1}}\:\Rightarrow\:{x}\:=\:\frac{{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}} \\ $$$$\int\:\frac{\sqrt{\frac{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{3}}{\mathrm{2}}−\mathrm{1}}}{{t}}×{t}\:{dt}\:=\:…
Question Number 20162 by ajfour last updated on 23/Aug/17 $${Compute}\:{the}\:{volume}\:{of}\:{a}\:{solid} \\ $$$${bounded}\:{by}\:{a}\:{surface}\:{with}\:{equation} \\ $$$$\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)^{\mathrm{2}} ={a}^{\mathrm{3}} {x}\:. \\ $$ Answered by ajfour last…
Question Number 151230 by mnjuly1970 last updated on 19/Aug/21 $$ \\ $$$$\:\:\:\:{prove}: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\:\left(\:\mathrm{1}+{x}^{\:\mathrm{2}} \right)}{{x}^{\:\mathrm{2}} \left(\mathrm{1}+{x}^{\:\mathrm{2}} \right)}{dx}=\:\pi\:{ln}\left(\frac{{e}}{\mathrm{2}}\:\right)\:.. \\ $$ Commented by Lordose last…