Question Number 129064 by Eric002 last updated on 12/Jan/21 $${valuate}\:{the}\:{following}\:{integral} \\ $$$${I}=\int_{\mathrm{1}} ^{\infty} \frac{{dt}}{\left({t}\right)^{\mathrm{2}{k}+{v}+\frac{\mathrm{3}}{\mathrm{2}}} \sqrt{{t}−\mathrm{1}}} \\ $$$${and}\:{prove}\:{that}: \\ $$$${I}=\sqrt{\pi}\frac{\Gamma\left({v}+\mathrm{1}\right)}{\Gamma\left({v}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\:\left(\frac{\left(\frac{{v}+\mathrm{1}}{\mathrm{2}}\right)_{{k}} \left(\mathrm{1}+\frac{{v}}{\mathrm{2}}\right)_{{k}} }{\left(\frac{{v}}{\mathrm{2}}+\frac{\mathrm{3}}{\mathrm{4}}\right)_{{k}} \left(\frac{{v}}{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{4}}\right)_{{k}} }\right) \\ $$…
Question Number 129060 by benjo_mathlover last updated on 12/Jan/21 $$\:\phi\:=\:\int\:\frac{\mathrm{ln}\:\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\: \\ $$ Answered by liberty last updated on 12/Jan/21 $$\:\mathrm{let}\:\mathrm{ln}\:\left(\mathrm{x}\right)=\mathrm{h}\:\Rightarrow\mathrm{x}\:=\:\mathrm{e}^{\mathrm{h}} \\ $$$$\:\phi\:=\:\int\:\frac{\mathrm{h}}{\mathrm{e}^{\mathrm{2h}} }\:\left(\mathrm{e}^{\mathrm{h}} \:\mathrm{dh}\:\right)=\:\int\:\mathrm{h}.\mathrm{e}^{−\mathrm{h}}…
Question Number 63519 by Rio Michael last updated on 05/Jul/19 $${consider}\:{the}\:{general}\:{definite}\:{intergral}\:\: \\ $$$$\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{{n}} {xdx} \\ $$$$\left.{a}\right)\:{prove}\:{that}\:{for}\:{n}\geqslant\mathrm{2},\:{nI}_{{n}} =\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} . \\ $$$$\left.{b}\left.\right)\left.\:{Find}\:{the}\:{values}\:{of}\:\:\boldsymbol{{i}}\right)\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{5}}…
Question Number 129057 by pipin last updated on 12/Jan/21 $$\: \\ $$$$\:\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\mathrm{dx}\:=\:… \\ $$ Answered by MJS_new last updated on 12/Jan/21 $$\int\frac{{dx}}{{x}^{\mathrm{3}}…
Question Number 63510 by turbo msup by abdo last updated on 05/Jul/19 $${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}−\mathrm{1}} }{{x}+{t}}\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$${and}\:\mathrm{0}<{a}<\mathrm{1} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}−\mathrm{1}} }{\left({x}+{t}\right)^{\mathrm{2}}…
Question Number 63508 by mathmax by abdo last updated on 05/Jul/19 $${let}\:\:{f}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{ixt}\:−\mathrm{1}\right)}\:\:{with}\:\mid{x}\mid>\mathrm{2}\:\:\:\left({i}^{\mathrm{2}} =−\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({f}\left({x}\right)\right)\:{and}\:{Im}\left({f}\left({x}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:\:{find}\:{olso}\:{g}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{t}}{\left({t}^{\mathrm{2}} \:+{ixt}\:−\mathrm{1}\right)^{\mathrm{2}}…
Question Number 63509 by mathmax by abdo last updated on 05/Jul/19 $${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$ Commented by Prithwish sen last updated on…
Question Number 129038 by bramlexs22 last updated on 12/Jan/21 $$\int\:\frac{\mathrm{5e}^{\mathrm{4t}} +\mathrm{10e}^{\mathrm{2t}} +\mathrm{2}}{\mathrm{e}^{\mathrm{2t}} +\mathrm{2}}\:\mathrm{dt}\: \\ $$ Answered by liberty last updated on 12/Jan/21 $$\:\int\:\frac{\left(\mathrm{5e}^{\mathrm{2t}} +\mathrm{1}\right)\left(\mathrm{e}^{\mathrm{2t}} +\mathrm{2}\right)−\mathrm{e}^{\mathrm{2t}}…
Question Number 129031 by oustmuchiya@gmail.com last updated on 12/Jan/21 $${Given}\:{that}\:\boldsymbol{{tan}}^{−\mathrm{1}} \boldsymbol{{x}}\:{show}\:{that}\:\: \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}\:=\:\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} } \\ $$ Answered by MJS_new last updated on 12/Jan/21 $${y}=\mathrm{arctan}\:{x} \\…
Question Number 129033 by oustmuchiya@gmail.com last updated on 12/Jan/21 $${prove}\:{using}\:{the}\:{first}\:{principle}\:{that} \\ $$$${the}\:{derivative}\:{of}\:\boldsymbol{{sin}}\:\boldsymbol{{x}}\:{is}\:\boldsymbol{{cox}}\:\boldsymbol{{x}}\:{and} \\ $$$${that}\:{the}\:{derivative}\:{of}\:\boldsymbol{{cos}}\:\boldsymbol{{x}}\:{is} \\ $$$$−\boldsymbol{{sinx}} \\ $$ Commented by bramlexs22 last updated on 12/Jan/21…