Category: Integration

Question Number 151246 by mathmax by abdo last updated on 19/Aug/21 $$\mathrm{find}\mathrm{I}={\int }_0^{\frac{\pi }{4}}\ln \left(\mathrm{cosx}\right)\mathrm{dx}\mathrm{and}\mathrm{J}={\int }_0^{\frac{\pi }{4}}\ln \left(\mathrm{sinx}\right)\mathrm{dx}$$ Answered by qaz last updated on 19/Aug/21…

Question Number 20167 by tammi last updated on 23/Aug/17 $$pleasesolveit$$$$integratewithrespecttox$$$$\int \frac{5x-2}{3x^2+2x+1}$$ 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 85701 by jagoll last updated on 24/Mar/20 $$\int \frac{\sqrt{3\mathrm{x}-1}}{\sqrt{2\mathrm{x}+1}}\mathrm{dx}$$ Commented by john santu last updated on 24/Mar/20 $$lett=\sqrt{2x+1}\Rightarrow x=\frac{t^2-1}{2}$$$$\int\:\frac{\sqrt{\frac{\mathrm{3}{t}^{\mathrm{2}} −\mathrm{3}}{\mathrm{2}}−\mathrm{1}}}{{t}}×{t}\:{dt}\:=\:…

Question Number 151230 by mnjuly1970 last updated on 19/Aug/21 $$$$$$prove:$$$${\int }_0^{\mathrm{\infty }}\frac{ln(1+x^2)}{x^2(1+x^2)}dx=\pi ln\left(\frac{e}{2}\right)..$$ Commented by Lordose last…

Question Number 151220 by peter frank last updated on 19/Aug/21 $$\mathrm{show}\mathrm{that}$$$${\int }_0^{\frac{\pi }{2}}\frac{\cos \mathrm{x}}{\cos \mathrm{x}+\sin \mathrm{x}+1}\mathrm{dx}=\frac{1}{4}(\pi -2\ln 2)$$ Answered by ArielVyny last updated on 19/Aug/21 $${t}={tan}\left(\frac{{x}}{\mathrm{2}}\right)\rightarrow{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+{t}^{\mathrm{2}}…