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Category: Integration

tan-x-a-btan-2-x-dx-

Question Number 38058 by ajfour last updated on 21/Jun/18 $$\int\frac{\mathrm{tan}\:{x}}{{a}+{b}\mathrm{tan}\:^{\mathrm{2}} {x}}\:{dx}\:\:=\:? \\ $$ Answered by behi83417@gmail.com last updated on 21/Jun/18 $${tg}^{\mathrm{2}} {x}=\frac{{a}}{{b}}{t}\Rightarrow\mathrm{2}{tgx}\left(\mathrm{1}+{tg}^{\mathrm{2}} {x}\right){dx}=\frac{{a}}{{b}}{dt} \\ $$$$\mathrm{2}{tgxdx}=\frac{\frac{{a}}{{b}}{dt}}{\mathrm{1}+\frac{{a}}{{b}}{t}}\Rightarrow{tgxdx}=\frac{{adt}}{\mathrm{2}\left({b}+{at}\right)}…

Prove-that-0-sin-x-e-x-1-dx-1-2-picoth-pi-1-solution-0-e-x-sin-x-1-e-x-dx-0-si

Question Number 169115 by mnjuly1970 last updated on 24/Apr/22 $$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Prove}\:\:\:\:\mathrm{that} \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left({x}\right)}{{e}^{\:{x}} \:−\mathrm{1}}\:{dx}\:\overset{?} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\:\left(\:\pi{coth}\left(\pi\right)\:−\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\:−−−\:\:{solution}\:−−− \\ $$$$\:\:\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty}…

Question-169078

Question Number 169078 by MikeH last updated on 23/Apr/22 Answered by mahdipoor last updated on 23/Apr/22 $$\alpha^{\mathrm{2}} ×\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }=\frac{\partial{u}}{\partial{t}}\:\Rightarrow \\ $$$$−\alpha^{\mathrm{2}} \left(\frac{{n}\pi}{{L}}\right)^{\mathrm{2}} {exp}\left(−\frac{{n}^{\mathrm{2}} \alpha^{\mathrm{2}}…

x-a-2-cosx-b-2-sinx-dx-

Question Number 103537 by TMSF last updated on 15/Jul/20 $$\int\frac{\mathrm{x}}{\left(\mathrm{a}^{\mathrm{2}} \mathrm{cosx}+\mathrm{b}^{\mathrm{2}} \mathrm{sinx}\right)}\mathrm{dx} \\ $$ Answered by mathmax by abdo last updated on 15/Jul/20 $$\mathrm{let}\:\mathrm{I}\:=\int\:\:\frac{\mathrm{xdx}}{\mathrm{a}^{\mathrm{2}} \mathrm{cosx}\:+\mathrm{b}^{\mathrm{2}}…

3x-1-4x-2x-2-2-dx-

Question Number 169073 by MikeH last updated on 23/Apr/22 $$\int\frac{\mathrm{3}{x}}{\left(\mathrm{1}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$ Answered by MJS_new last updated on 24/Apr/22 $$\int\frac{\mathrm{3}{x}}{\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{1}\right)^{\mathrm{2}} }{dx}= \\…

3x-1-4x-x-2-dx-

Question Number 169054 by MikeH last updated on 23/Apr/22 $$\int\frac{\mathrm{3}{x}}{\left(\mathrm{1}−\mathrm{4}{x}−{x}^{\mathrm{2}} \right)}\:{dx} \\ $$ Answered by haladu last updated on 23/Apr/22 $$\:\: \\ $$$$\:\:\mathrm{1}−\mathrm{4}\boldsymbol{\mathrm{x}}\:−\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:\:=\:\:\mathrm{1}\:\:−\:\left(\:\boldsymbol{\mathrm{x}}\:+\mathrm{2}\:\right)^{\mathrm{2}} −\mathrm{4}\:…