Question Number 73715 by Learner-123 last updated on 15/Nov/19 $${Evaluate}\:{the}\:{integral}\:: \\ $$$$\underset{\:\mathbb{R}} {\int}\int\left(\mathrm{3}{x}^{\mathrm{2}} +\mathrm{14}{xy}+\mathrm{8}{y}^{\mathrm{2}} \right){dxdy}\:{for}\:{the}\:{region} \\ $$$$\mathbb{R}\:\mathrm{in}\:{the}\:\mathrm{1}{st}\:{quadrant}\:{bounded}\:{by}\:{the} \\ $$$${lines}\:{y}=\frac{−\mathrm{3}}{\mathrm{2}}{x}+\mathrm{1},{y}=\frac{−\mathrm{3}}{\mathrm{2}}{x}+\mathrm{3},{y}=−\frac{\mathrm{1}}{\mathrm{4}}{x} \\ $$$${and}\:{y}=−\frac{\mathrm{1}}{\mathrm{4}}{x}+\mathrm{1}\:. \\ $$ Commented by…
Question Number 139248 by ajfour last updated on 24/Apr/21 $$\int_{{a}} ^{\:{b}} {f}\left({t}\right){g}'\left({t}\right){dt}=\:? \\ $$ Answered by mathmax by abdo last updated on 25/Apr/21 $$\int_{\mathrm{a}} ^{\mathrm{b}}…
Question Number 139224 by mnjuly1970 last updated on 24/Apr/21 $$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)}{{x}}{dx} \\ $$$$\:\:\:\phi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{3}} \right)−{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$\:\:\:\:=\Omega+\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\:\:\:\:\Omega=−\Sigma\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{x}^{\mathrm{3}{n}−\mathrm{1}}…
Question Number 73689 by ajfour last updated on 14/Nov/19 $$\int_{−\mathrm{1}} ^{\:\:\mathrm{1}} \left(\mathrm{2}+{x}\right)\mathrm{sin}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{3}−\mathrm{3}{x}^{\mathrm{2}} }}{\mathrm{2}+{x}}\right){dx}\:=\:? \\ $$ Answered by mind is power last updated on 15/Nov/19…
Question Number 139217 by mnjuly1970 last updated on 24/Apr/21 $$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:…..\:{nice}\:….\:….\:{math}…. \\ $$$$\:\:\:{prove}\:{that}: \\ $$$$\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{ln}\left(\mathrm{1}+{sin}\left({x}\right).{cos}\left({x}\right)\right)}{{tan}\left({x}\right)}{dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{72}} \\ $$ Commented by liki last…
Question Number 8114 by uchechukwu okorie favour last updated on 30/Sep/16 $${Evaluate}\:\int\mathrm{cos}\:^{\mathrm{6}} {x} \\ $$ Answered by sandy_suhendra last updated on 30/Sep/16 Answered by prakash…
Question Number 8115 by uchechukwu okorie favour last updated on 30/Sep/16 $${if}\:{xy}+{y}^{\mathrm{2}} =\mathrm{1}.\:{Find}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}}\:{at}\:\left(\mathrm{0},\mathrm{1}\right) \\ $$ Answered by prakash jain last updated on 30/Sep/16 $${x}\frac{{dy}}{{dx}}+{y}+\mathrm{2}{y}\frac{{dy}}{{dx}}=\mathrm{0}…\left({i}\right)…
Question Number 139182 by aupo14 last updated on 23/Apr/21 Commented by mr W last updated on 23/Apr/21 $${there}\:{is}\:{not}\:{much}\:{to}\:{explain}.\:{it}\:{is} \\ $$$${just}\:{the}\:{definition}\:{of}\:{a}\:{special} \\ $$$${function}. \\ $$ Commented…
Question Number 139164 by mathlove last updated on 23/Apr/21 Answered by qaz last updated on 25/Apr/21 $${ln}\left(\mathrm{1}−{ae}^{{ix}} \right) \\ $$$$={ln}\left[\mathrm{1}−{a}\left(\mathrm{cos}\:{x}+{i}\mathrm{sin}\:{x}\right)\right] \\ $$$$={ln}\left(\sqrt{\left(\mathrm{1}−{a}\mathrm{cos}\:{x}\right)^{\mathrm{2}} +\left({a}\mathrm{sin}\:{x}\right)^{\mathrm{2}} }{e}^{{i}\mathrm{tan}^{−\mathrm{1}} \frac{−{a}\mathrm{sin}\:{x}}{\mathrm{1}−{a}\mathrm{cos}\:{x}}}…
Question Number 139153 by henderson last updated on 23/Apr/21 $$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{everybody}}\:! \\ $$$$\boldsymbol{\mathrm{for}}\:{f}\left({x}\right)=\int_{{x}} ^{\:{x}^{\mathrm{2}} } \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}}\:{dt} \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{domain}}\:\boldsymbol{\mathrm{of}}\:{f},\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:{f}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{even}}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:{f}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{differentiable}}\:\boldsymbol{\mathrm{on}}\:\mathbb{R},\:\boldsymbol{\mathrm{find}}\:{f}\:^{'} \left({x}\right). \\ $$$$\mathrm{3}.\:\boldsymbol{\mathrm{determine}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{limited}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{order}}\:\mathrm{4}\:\boldsymbol{\mathrm{of}}\:{f} \\ $$$$\boldsymbol{\mathrm{in}}\:\mathrm{0}.…