Category: Arithmetic

Question Number 149718 by peter frank last updated on 06/Aug/21 Answered by Ar Brandon last updated on 06/Aug/21 $$I=\int \cos 2\theta \ln \left(\frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }\right)d\theta$$$$=\int \cos 2\theta \ln \left(\frac{1+\tan \theta }{1-\tan \theta }\right)d\theta =\int \cos 2\theta \ln \left(\tan (\frac{\pi }{4}+\theta )\right)d\theta$$$$\{\begin{array}{l}u\left(x\right)=\ln \left(\tan (\frac{\pi }{4}+\theta )\right)\\ v^{\prime }\left(x\right)=\cos 2\theta \end{array}\Rightarrow \{\begin{array}{l}{u}^{\prime }\left(x\right)=\frac{2}{\sin (\frac{\pi }{2}+2\theta )}\\ v\left(x\right)=\frac{1}{2}\sin 2\theta \end{array}$$$${I}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin2}\theta\centerdot\mathrm{ln}\left(\mathrm{tan}\left(\frac{\pi}{\mathrm{4}}+\theta\right)\right)−\int\frac{\mathrm{sin2}\theta}{\mathrm{cos2}\theta}{d}\theta…

Question Number 83876 by M±th+et£s last updated on 07/Mar/20 Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 18213 by chux last updated on 16/Jul/17 $$\mathrm{The}\mathrm{greatest}\mathrm{number}\mathrm{that}\mathrm{will}$$$$\mathrm{divide}82,111\mathrm{and}140\mathrm{leaving}\mathrm{the}$$$$\mathrm{same}\mathrm{remainder}\mathrm{in}\mathrm{each}\mathrm{case}\mathrm{is}\dots \dots ..$$ Answered by Tinkutara last updated on 17/Jul/17 $$\mathrm{82}\:=\:{xa}\:+\:{r} \…

Question Number 83629 by M±th+et£s last updated on 04/Mar/20 $$showthat$$$$\underset{n,k=0}{\sum ^{\mathrm{\infty }}}\frac{n!k!}{(n+k+2)!}=\frac{{\pi }^2}{6}$$ Answered by Kamel Kamel last updated on 04/Mar/20…

Question Number 17767 by Joel577 last updated on 10/Jul/17 $$\underset{n=1}{\sum ^{30}}(n^2+1)=$$$$\left(\mathrm{A}\right)\underset{n=1}{\sum ^{15}}(2n^2+30n+224)$$$$\left(\mathrm{B}\right)\underset{n=1}{\sum ^{15}}(2n^2+30n+225)$$$$\left(\mathrm{C}\right)\:\underset{{n}\:=\:\mathrm{1}}…