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Category: Integration

solve-for-x-R-0-sin-xt-e-t-1-dt-2-coth-x-1-2x-

Question Number 218563 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:{solve}\:{for}\:\boldsymbol{{x}}\:\in\:\mathbb{R}\: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}\left({xt}\right)}{{e}^{{t}} −\mathrm{1}\:}\:{dt}\:=\:\frac{\boldsymbol{\pi}}{\mathrm{2}}\:{coth}\left(\boldsymbol{\pi}{x}\right)\:−\:\frac{\mathrm{1}}{\mathrm{2}{x}}\:\: \\ $$$$ \\ $$ Terms of Service Privacy…

A-kind-of-calculation-relatedt-o-arctangent-integral-Exere-0-1-arctan-2-x-1-x-2-dx-Solution-0-1-arctan-2-x-x-dx-A-0-1-x-arctan-2-x-1-x-2-dx-B-0-1-x-arctan-2-

Question Number 218594 by MrGaster last updated on 12/Apr/25 $$\mathrm{A}\:\mathrm{kind}\:\mathrm{of}\:\mathrm{calculation}\:\mathrm{relatedt} \\ $$$$\mathrm{o}\:\mathrm{arctangent}\:\mathrm{integral}: \\ $$$$\boldsymbol{\mathrm{Exere}}:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arctan}^{\mathrm{2}} }{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}. \\ $$$$\mathrm{Solution}:\underset{{A}} {\underbrace{=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arctan}^{\mathrm{2}} {x}}{{x}}{dx}}}\:−\underset{{B}} {\underbrace{\int_{\mathrm{0}}…

dx-2x-x-2-3-

Question Number 218462 by York12 last updated on 10/Apr/25 $$\int\frac{{dx}}{\:\sqrt{\mathrm{2}{x}−{x}^{\mathrm{2}} +\mathrm{3}}} \\ $$ Answered by som(math1967) last updated on 10/Apr/25 $$\int\frac{{dx}}{\:\sqrt{\mathrm{4}−\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}\right)}} \\ $$$$=\int\frac{{dx}}{\:\sqrt{\mathrm{2}^{\mathrm{2}} −\left({x}−\mathrm{1}\right)^{\mathrm{2}}…

0-J-ar-r-2-k-2-dr-

Question Number 218376 by Nicholas666 last updated on 08/Apr/25 $$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{J}}_{\boldsymbol{\alpha}} \left(\boldsymbol{{ar}}\right)}{\left(\boldsymbol{{r}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} \right)\boldsymbol{\mu}}\boldsymbol{{dr}}\: \\ $$$$ \\ $$ Terms of Service Privacy…

tan-x-sin-3-x-cos-x-dx-

Question Number 218375 by Nicholas666 last updated on 08/Apr/25 $$ \\ $$$$\:\:\int\frac{\sqrt{\boldsymbol{{tan}}\:\boldsymbol{{x}}}}{\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{x}}\:\boldsymbol{{cos}}\:\boldsymbol{{x}}}\boldsymbol{{dx}} \\ $$$$ \\ $$ Answered by Ghisom last updated on 08/Apr/25 $$\int\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{sin}^{\mathrm{3}}…