Category: Integration

Question Number 128285 by john_santu last updated on 06/Jan/21 $$\cos \left(\frac{\pi }{7}\right)-\cos \left(\frac{2\pi }{7}\right)+\cos \left(\frac{3\pi }{7}\right)=?$$ Answered by liberty last updated on 06/Jan/21 $$\mathrm{let}\phi =\cos \left(\frac{\pi }{7}\right)-\cos \left(\frac{2\pi }{7}\right)+\cos \left(\frac{3\pi }{7}\right)$$$$\phi =\cos \left(\frac{\pi }{7}\right)+\cos \left(\frac{3\pi }{7}\right)+\cos \left(\frac{5\pi }{7}\right)$$$$\:\Rightarrow\mathrm{2}\varphi\mathrm{sin}\:\mathrm{t}\:=\:\mathrm{2sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{t}\:+\:\mathrm{2sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{3t}+\mathrm{2sin}\:\mathrm{tcos}\:\mathrm{5t} \…

Question Number 62732 by mathmax by abdo last updated on 24/Jun/19 $$calculate{\int }_0^{2\pi }\frac{cos\left(2x\right)}{2cosx-sin\left(x\right)}dx$$ Answered by MJS last updated on 24/Jun/19 $$\frac{\mathrm{cos}\:\left(\mathrm{2}\left({x}+\pi\right)\right)}{\mathrm{2cos}\:\left({x}+\pi\right)\:−\mathrm{sin}\:\left({x}+\pi\right)}=−\frac{\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{2cos}\:{x}\:−\mathrm{sin}\:{x}}\:\Rightarrow \…

Question Number 128262 by Ahmed1hamouda last updated on 06/Jan/21 Answered by mr W last updated on 06/Jan/21 $$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{{dxdy}}{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}}…

Question Number 128244 by mnjuly1970 last updated on 05/Jan/21 $$\dots nicecalculus\dots$$$$provethat::\mathrm{\Omega }={\int }_0^{\frac{\pi }{4}}ln\left(sin\left(x\right)\right)d=\frac{-\pi }{4}log\left(2\right)-\frac{G}{2}$$$$log\left(2sin\left(x\right)\right)=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{-1}{n}cos\left(2nx\right)$$$$\:\Omega=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left\{−{log}\left(\mathrm{2}\right)−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cos}\left(\mathrm{2}{nx}\right)}{{n}}\right\}{dx} \…

Question Number 128194 by Algoritm last updated on 05/Jan/21 Commented by MJS_new last updated on 05/Jan/21 $${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{possible}\mathrm{by}\mathrm{using}$$$$\cos x=\frac{{\mathrm{e}}^{\mathrm{i}x}+{\mathrm{e}}^{-\mathrm{i}x}}{2}=\frac{{\mathrm{e}}^{2\mathrm{i}x}+1}{{2\mathrm{e}}^{\mathrm{i}x}}\Rightarrow$$$$\mathrm{8}\int\frac{{x}\mathrm{e}^{\mathrm{3i}{x}}…