Question Number 115558 by mnjuly1970 last updated on 26/Sep/20 $$\:\:\:\:\:\:\:…\:{advanced}\:\:\:{calculus}…\: \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\mathrm{1}+{ax}^{\mathrm{2}} \right){ln}\left(\mathrm{1}+\frac{{b}}{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\:\:\:\:\:\:\:{m}.{n}.{july} \\ $$$$ \\…
Question Number 50007 by Joel578 last updated on 13/Dec/18 $$\mathrm{For}\:{a}\:<\:{x}\:<\:{b},\:\mathrm{find}\: \\ $$$$\underset{{a}} {\overset{{b}} {\int}}\:\sqrt{{x}−{a}}\:.\:\sqrt{{b}−{x}}\:{dx} \\ $$ Commented by Abdo msup. last updated on 13/Dec/18 $${changement}\:\sqrt{{x}−{a}}={t}\:{give}\:{x}−{a}={t}^{\mathrm{2}}…
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Question Number 115507 by mnjuly1970 last updated on 26/Sep/20 $$\:\:\:\:\:\:\:\:\:\:\:….\:\:\:…{matematical}\:{analysis}…\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}>\mathrm{0}\:::\:\:\:\begin{bmatrix}{{i}\::\:\:\int_{\mathrm{0}\:} ^{\:\infty} \frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:{dx}=\:\sqrt{\pi{a}}}\\{{ii}:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({ax}\right)}{\:\sqrt{{x}}}\:{dx}\:=\:\frac{−\mathrm{1}+\mathrm{3}\sqrt{\mathrm{3}\:}}{\mathrm{4}}\:\sqrt{\frac{\pi}{\mathrm{6}{a}}\:\:}\:}\end{bmatrix} \\ $$$$\:\:\:\:\:…
Question Number 49967 by maxmathsup by imad last updated on 12/Dec/18 $$\:\:{calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −{i}\right)^{\mathrm{2}} } \\ $$ Commented by Abdo msup. last updated on…
Question Number 49968 by maxmathsup by imad last updated on 12/Dec/18 $${find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{t}\:{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{4}} }\:{dt} \\ $$ Commented by mathmax by abdo last updated on…
Question Number 49956 by maxmathsup by imad last updated on 12/Dec/18 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left({n}\:{arcosx}\right){dx}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$ Commented by Abdo msup. last updated on 13/Dec/18 $${we}\:{have}\:{cos}\left({n}\:{arcosx}\right)\:+{i}\:{sin}\left({narcosx}\right)…
Question Number 49954 by maxmathsup by imad last updated on 12/Dec/18 $${find}\:\:\:{f}\left(\alpha\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+\alpha^{\mathrm{2}} {x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }\:\:{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\mathrm{3}{x}\right)}{\mathrm{1}+\mathrm{9}{x}^{\mathrm{2}} }\:{dx}\:. \\…