Category: Integration

Question Number 156869 by cortano last updated on 16/Oct/21 Answered by gsk2684 last updated on 16/Oct/21 $$\int \frac{\sec x\left(\sec x(\sec x+\tan x)\right)}{{(\sec x+\tan x)}^6}dx$$$$put\sec x+\tan x=t,\sec x-\tan x=\frac{1}{t}\Rightarrow$$$$\sec x\tan x+\sec {}^{2}x=\frac{dt}{dx},2\sec x=t+\frac{1}{t}$$$$\int\frac{{t}+\frac{\mathrm{1}}{{t}}}{\mathrm{2}{t}^{\mathrm{6}}…

Question Number 156860 by amin96 last updated on 16/Oct/21 $${\int }_0^{\mathrm{\infty }}\frac{\sqrt[n]{x}}{x^3+x^2+x+1}dx=?$$ Answered by mindispower last updated on 16/Oct/21 $${f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty}…

Question Number 91310 by  M±th+et+s last updated on 29/Apr/20 $$showthat$$$${\int }_0^{\frac{\pi }{2}}cot\left(x\right)ln\left(sec\left(x\right)\right)dx=\frac{{\pi }^2}{24}$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 156841 by mnjuly1970 last updated on 16/Oct/21 Answered by gsk2684 last updated on 16/Oct/21 $$let\mathit{C}=\cos \frac{\pi }{2n+1}.\cos \frac{2\pi }{2n+1}.\cos \frac{3\pi }{2n+1}\dots \cos \frac{(n-1)\pi }{2n+1}.\cos \frac{n\pi }{2n+1}$$$$letS=\sin \frac{\pi }{2n+1}.\sin \frac{2\pi }{2n+1}.\sin \frac{3\pi }{2n+1}\dots \sin \frac{(n-1)\pi }{2n+1}.\sin \frac{n\pi }{2n+1}$$$$2^n.S.C=\left(2\sin \frac{\pi }{2n+1}\cos \frac{\pi }{2n+1}\right)\left(2\sin \frac{2\pi }{2n+1}\cos \frac{2\pi }{2n+1}\right)\left(2\sin \frac{3\pi }{2n+1}\cos \frac{3\pi }{2n+1}\right)\dots \left(2\sin \frac{(n-1)\pi }{2n+1}\cos \frac{(n-1)\pi }{2n+1}\right)\left(2\sin \frac{n\pi }{2n+1}\cos \frac{n\pi }{2n+1}\right)$$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{2}{n}+\mathrm{1}}.\mathrm{sin}\:\frac{\mathrm{4}\pi}{\mathrm{2}{n}+\mathrm{1}}.\mathrm{sin}\:\frac{\mathrm{6}\pi}{\mathrm{2}{n}+\mathrm{1}}…\mathrm{sin}\:\frac{\mathrm{2}\left({n}−\mathrm{1}\right)\pi}{\mathrm{2}{n}+\mathrm{1}}.\mathrm{sin}\frac{\mathrm{2}{n}\pi}{\mathrm{2}{n}+\mathrm{1}} \…

Question Number 25769 by abdo imad last updated on 14/Dec/17 $${a}−{nser}\:{to}\:{question}\:\mathrm{25765}…{we}\:{put}\:{I}=\int_{\mathrm{0}} ^{\infty} \left({cos}\left({x}^{\mathrm{2}{n}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{−\mathrm{1}} {dx}\:{and}\:{J}=\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{\mathrm{2}{n}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{−\mathrm{1}} {dx}…{we}\:{have}\:\mathrm{2}\left({I}+{iJ}\right)=\int_{{R}} {e}^{{ix}^{\mathrm{2}{n}} } \left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{−\mathrm{1}}…