Question Number 2284 by Yozzi last updated on 13/Nov/15 $${Define}\:{a}\:{curve}\:{E}\:{by}\:{the}\:{parametric} \\ $$$${equations} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\left({t}\right)=\int_{{g}_{\mathrm{1}} \left({t}\right)} ^{{h}_{\mathrm{1}} \left({t}\right)} {f}_{\mathrm{1}} \left({u}\right){du} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\left({t}\right)=\int_{{g}_{\mathrm{2}} \left({t}\right)} ^{{h}_{\mathrm{2}} \left({t}\right)} {f}_{\mathrm{2}}…
Question Number 67799 by mathmax by abdo last updated on 31/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:−{e}^{{ia}} \right)\left({x}^{\mathrm{2}} −{e}^{{ib}} \right)}\:\:{with}\:{a}>\mathrm{0}\:{andb}>\mathrm{0} \\ $$ Commented by MJS last updated…
Question Number 2241 by Yozzi last updated on 10/Nov/15 $${Evaluate} \\ $$$$\:\:\:\:\:\:\:\:\frac{\mathrm{9}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \left(\frac{{sin}\theta{cos}\theta}{{cos}^{\mathrm{3}} \theta+{sin}^{\mathrm{3}} \theta}\right)^{\mathrm{2}} {d}\theta. \\ $$ Answered by Filup last updated on…
Question Number 133305 by liberty last updated on 21/Feb/21 $$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\frac{\mathrm{dx}}{\mathrm{5}+\mathrm{3sin}\:\mathrm{2x}}\:=? \\ $$ Answered by physicstutes last updated on 21/Feb/21 $$\mathrm{set}\:{t}\:=\:\mathrm{tan}\:{x} \\ $$$$\Rightarrow\:\mathrm{sin}\:\mathrm{2}{x}\:=\:\mathrm{2}\:\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}\:=\:\frac{\mathrm{2}\:\mathrm{tan}\:{x}}{\mathrm{1}+\:\mathrm{tan}^{\mathrm{2}} {x}}\:\:=\:\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}}…
Question Number 2212 by Yozzi last updated on 09/Nov/15 $$\int\frac{{dt}}{\left(\mathrm{1}−{kt}\right)\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}=?\:\:\mathrm{0}<{k}<\mathrm{1} \\ $$ Commented by 123456 last updated on 09/Nov/15 $$\frac{\mathrm{ln}\:\left(\sqrt{{k}^{\mathrm{2}} −\mathrm{1}}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+{k}−{x}\right)−\mathrm{ln}\:\left(\mathrm{1}−{kx}\right)}{\:\sqrt{{k}^{\mathrm{2}} −\mathrm{1}}} \\…
Question Number 133291 by liberty last updated on 21/Feb/21 $$\:\int\:\frac{\left(\mathrm{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{6}} \right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$ Answered by EDWIN88 last updated on 21/Feb/21 $$\mathrm{integration}\:\mathrm{by}\:\mathrm{parts} \\…
Question Number 2210 by Yozzi last updated on 08/Nov/15 $${Evaluate}\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} {cos}\alpha+\mathrm{1}}\:\:\left(\mathrm{0}<\alpha<\pi\right). \\ $$ Commented by 123456 last updated on 09/Nov/15…
Question Number 67744 by mathmax by abdo last updated on 31/Aug/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({t}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({t}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}}…
Question Number 133268 by mnjuly1970 last updated on 20/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:….{calculus}… \\ $$$$\:\:{prove}:: \\ $$$$\:\:\:\boldsymbol{\phi}=\int_{−\infty} ^{\:+\infty} \frac{{dx}}{\left({x}^{\mathrm{2}} +\pi^{\mathrm{2}} \right){cosh}\left({x}\right)}=\frac{\mathrm{4}}{\boldsymbol{\pi}}\:−\mathrm{1} \\ $$$$ \\ $$ Answered by Ajetunmobi…
Question Number 2186 by Yozzi last updated on 07/Nov/15 $${Let}\:{J}=\int_{\mathrm{0}} ^{\infty} {f}\left(\left({x}−{x}^{−\mathrm{1}} \right)^{\mathrm{2}} \right){dx}\:{where}\:{f}\:{is} \\ $$$${any}\:{function}\:{for}\:{which}\:{the}\:{integral} \\ $$$${exists}.\:{Show}\:{that} \\ $$$${J}=\int_{\mathrm{0}} ^{\infty} {x}^{−\mathrm{2}} {f}\left(\left({x}−{x}^{−\mathrm{1}} \right)^{\mathrm{2}} \right){dx}=\mathrm{0}.\mathrm{5}\int_{\mathrm{0}}…