Question Number 66060 by mathmax by abdo last updated on 08/Aug/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{{x}+{tant}}\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{aexplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left({x}+{tant}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right){at}\:{form}\:{of}\:{integral} \\…
Question Number 66061 by mathmax by abdo last updated on 08/Aug/19 $${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sinx}}{\mathrm{1}+{te}^{−{x}^{\mathrm{2}} } }{dx}\:\:\:\:{with}\:\mid{t}\mid<\mathrm{1} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$ Terms of Service Privacy Policy…
Question Number 66058 by arvinddayama01@gmail.comm last updated on 08/Aug/19 $${If}\:\:\:\:{x}\:+\:\frac{\mathrm{1}}{{x}}=\mathrm{1} \\ $$$$ \\ $$$${find}\:{out}\:{value}:− \\ $$$$ \\ $$$$\:\:\:\:\frac{{x}^{\mathrm{20}} +{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} }{{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} +{x}^{\mathrm{8}}…
Question Number 131595 by Algoritm last updated on 06/Feb/21 Commented by Algoritm last updated on 06/Feb/21 $$\mathrm{x}_{\mathrm{1}} =\mathrm{37}/\mathrm{2}\:\:\:\mathrm{x}_{\mathrm{2}} =\mathrm{77}/\mathrm{4}\:\:\:\mathrm{x}_{\mathrm{3}} =\mathrm{65}/\mathrm{4} \\ $$$$\Sigma\boldsymbol{\mathrm{x}}=\mathrm{37}/\mathrm{2}+\mathrm{77}/\mathrm{4}+\mathrm{65}/\mathrm{4}=\mathrm{54}\:\: \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}! \\…
Question Number 519 by Yugi last updated on 25/Jan/15 $${Find}\:{the}\:{sum}\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(\mathrm{3}{r}+\mathrm{5}\right)\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix}=\mathrm{5}\begin{pmatrix}{{n}}\\{\mathrm{0}}\end{pmatrix}\:+\mathrm{8}\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\:+\mathrm{11}\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\:+…\left(\mathrm{3}{n}+\mathrm{5}\right)\begin{pmatrix}{{n}}\\{{n}}\end{pmatrix}\: \\ $$$${as}\:{a}\:{simple}\:{function}\:{of}\:{n}. \\ $$ Commented by prakash jain last updated on 22/Jan/15 $$\underset{{r}=\mathrm{0}}…
Question Number 66055 by ajfour last updated on 08/Aug/19 Commented by ajfour last updated on 08/Aug/19 $${Eq}.\:{of}\:{ellipse}\:\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{\left({y}−{b}\right)^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$${parabola}\:{touches}\:{the}\:{x}-{axis} \\ $$$${and}\:{the}\:{ellipse}\:{and}\:{passes}…
Question Number 518 by Yugi last updated on 25/Jan/15 $${Give}\:{the}\:{result}\:{of}\:{the}\:{following}\:{computation}\:{as}\:{an}\:{integer}\:{in}\:{the}\:{usual}\:{decimal} \\ $$$${form}.\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{303},\mathrm{000},\mathrm{000},\mathrm{000},\mathrm{303}×\mathrm{3},\mathrm{300},\mathrm{000},\mathrm{033}}{\mathrm{1},\mathrm{000},\mathrm{100},\mathrm{010},\mathrm{001}} \\ $$ Answered by prakash jain last updated on 22/Jan/15 $$\frac{\mathrm{303}\left(\mathrm{10}^{\mathrm{12}}…
Question Number 517 by Yugi last updated on 25/Jan/15 $${A}\:{person}\:{is}\:{said}\:{to}\:{be}\:{n}\:{years}\:{old}\:\left(\:{where}\:{n}\:{is}\:{a}\:{non}−{negative}\:{integer}\right)\:{if}\: \\ $$$${the}\:{person}\:{has}\:{lived}\:{at}\:{least}\:{n}\:{years}\:{and}\:{has}\:{not}\:{lived}\:{n}+\mathrm{1}\:{years}.\:{At}\:{some}\:{point} \\ $$$${Tom}\:{is}\:\mathrm{4}\:{years}\:{old}\:{and}\:{John}\:{is}\:{three}\:{times}\:{as}\:{old}\:{as}\:{Mary}.\:{At}\:{another}\:{time}, \\ $$$${Mary}\:{is}\:{twice}\:{as}\:{old}\:{as}\:{Tom}\:{and}\:{John}\:{is}\:{five}\:{times}\:{as}\:{old}\:{as}\:{Tom}.\:{At}\:{a}\:{third}\: \\ $$$${time},\:{John}\:{is}\:{twice}\:{as}\:{old}\:{as}\:{Mary}\:{and}\:{Tom}\:{is}\:{t}\:{years}\:{old}.\:{What}\:{is}\:{the}\:{largest} \\ $$$${possible}\:{value}\:{of}\:{t}? \\ $$ Commented by prakash…
Question Number 131590 by pticantor last updated on 06/Feb/21 $$\boldsymbol{{show}}\:\boldsymbol{{that}} \\ $$$$\boldsymbol{{U}}_{\boldsymbol{{n}}} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{{dx}}}{\mathrm{1}+\boldsymbol{{x}}+\boldsymbol{{x}}^{\mathrm{2}} +….+\boldsymbol{{x}}^{\boldsymbol{{n}}} }\: \\ $$$$\boldsymbol{{converges}}!! \\ $$ Answered by mathmax by…
Question Number 516 by 112358 last updated on 25/Jan/15 $${Find}\:{all}\:{triangles}\:{with}\:{consecutive} \\ $$$${integer}\:{sides}\:{and}\:{having}\:{an}\:{angle}\:{twice} \\ $$$${another}\:{angle}. \\ $$ Commented by prakash jain last updated on 22/Jan/15 $${a}={n}…