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Category: Integration

cos-7x-cos-8x-1-2cos-5x-dx-

Question Number 136992 by liberty last updated on 28/Mar/21 $$\ell\:=\:\int\:\frac{\mathrm{cos}\:\mathrm{7x}−\mathrm{cos}\:\mathrm{8x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{5x}}\:\mathrm{dx}\:=? \\ $$ Answered by EDWIN88 last updated on 28/Mar/21 $$\:\ell\:=\:\int\:\frac{\mathrm{cos}\:\mathrm{7x}−\mathrm{cos}\:\mathrm{8x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{5x}}\:\mathrm{dx} \\ $$$$\mathrm{Simplify}\:\mathrm{the}\:\mathrm{integrand}\:\mathrm{by}\:\mathrm{Euler}'\mathrm{s}\:\mathrm{formula} \\ $$$$\frac{\mathrm{cos}\:\mathrm{7x}−\mathrm{cos}\:\mathrm{8x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{5x}}\:=\:\frac{\mathrm{e}^{\mathrm{i7x}} +\mathrm{e}^{−\mathrm{i7x}}…

1-8-x-2-4-x-2-dx-2-x-6-2x-2-5x-3-

Question Number 136968 by Eric002 last updated on 28/Mar/21 $$\left.\mathrm{1}\right)\int\frac{\mathrm{8}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}−{x}^{\mathrm{2}} }}{dx} \\ $$$$\left.\mathrm{2}\right)\int\frac{{x}−\mathrm{6}\:}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{3}} \\ $$ Answered by EDWIN88 last updated on 28/Mar/21 $$\left(\mathrm{1}\right)\int\:\frac{\mathrm{8}}{\mathrm{x}^{\mathrm{3}}…

0-pi-2-1-sec-2-t-sec-t-1-sec-t-2-2-dt-

Question Number 136969 by liberty last updated on 28/Mar/21 $$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \frac{\left(\mathrm{1}+\mathrm{sec}\:^{\mathrm{2}} \mathrm{t}\right)\:\sqrt{\mathrm{sec}\:\mathrm{t}}}{\left(\mathrm{1}+\mathrm{sec}\:\mathrm{t}\right)^{\mathrm{2}} −\mathrm{2}}\:\mathrm{dt}\:=?\: \\ $$ Answered by EDWIN88 last updated on 28/Mar/21 $$\:\mathrm{E}\:=\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}}…

csc-4-x-cot-2-x-dx-

Question Number 136971 by Eric002 last updated on 28/Mar/21 $$\int{csc}^{\mathrm{4}} \left({x}\right)\:{cot}^{\mathrm{2}} \left({x}\right)\:{dx} \\ $$ Answered by Ñï= last updated on 28/Mar/21 $$\int{csc}\:^{\mathrm{4}} {x}\mathrm{cot}\:^{\mathrm{2}} {xdx}=−\int\left(\mathrm{1}+\mathrm{cot}\:^{\mathrm{2}} {x}\right)\mathrm{cot}\:^{\mathrm{2}}…

4-x-2-x-dx-

Question Number 136948 by leena12345 last updated on 28/Mar/21 $$\int\frac{\sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}{{x}}{dx} \\ $$ Answered by Olaf last updated on 28/Mar/21 $$\mathrm{F}\left({x}\right)\:=\:\int\frac{\sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}{{x}}\:{dx} \\ $$$$\mathrm{F}\left({u}\right)\:\overset{{x}=\mathrm{2sh}{u}} {=}\:\:\:\:\int\frac{\sqrt{\mathrm{4}+\mathrm{4sh}^{\mathrm{2}}…