Question Number 129150 by benjo_mathlover last updated on 13/Jan/21 $$\:\int_{\mathrm{0}} ^{\:\mathrm{x}} \left(\mathrm{x}−\mathrm{u}\right)^{\mathrm{2}} \mathrm{u}\:\mathrm{f}\left(\mathrm{u}\right)\:\mathrm{du}\:=\:\mathrm{4x}−\mathrm{6sin}\:\mathrm{x}+\mathrm{2xcos}\:\mathrm{x} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=? \\ $$ Answered by liberty last updated on 13/Jan/21 $$\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{0}}…
Question Number 63570 by mr W last updated on 05/Jul/19 Commented by mathmax by abdo last updated on 05/Jul/19 $$\int_{\mathrm{0}} ^{\mathrm{1}} \left(−\mathrm{1}\right)^{{x}} {dx}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{i}\pi{x}}…
Question Number 63566 by aliesam last updated on 05/Jul/19 $${prove}\:{that} \\ $$$$ \\ $$$$\int{sin}^{{n}} \left({x}\right)\:{dx}\:,\:{p}\in{n}\:,\:{p}\geqslant\mathrm{2}\:=−\:\frac{\mathrm{1}}{{n}}{cos}\left({x}\right)\:{sin}^{{n}−\mathrm{1}} \left({x}\right)\:+\:\left({p}−\mathrm{1}\right)\int{sin}^{{n}−\mathrm{2}} \left({x}\right)\:{dx} \\ $$ Commented by aliesam last updated on…
Question Number 129068 by bagjagunawan last updated on 12/Jan/21 Answered by mnjuly1970 last updated on 12/Jan/21 $$\:\:{ans}\:::\:\:\pi{ln}\left(\mathrm{1}−{e}^{−\pi{n}} \right)\:…. \\ $$ Terms of Service Privacy Policy…
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}}…