Question Number 206754 by mathzup last updated on 23/Apr/24 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−\sqrt{{x}}}{ln}^{\mathrm{2}} \left({x}\right){dx} \\ $$ Answered by Berbere last updated on 24/Apr/24 $${x}={u}^{\mathrm{2}} \Rightarrow{dx}=\mathrm{2}{udu} \\…
Question Number 206721 by ajfour last updated on 23/Apr/24 $$\int\frac{{xdx}}{{x}+\mathrm{4}}=?\:\:\:\:\:\:\:{please} \\ $$ Answered by A5T last updated on 23/Apr/24 $$\frac{{x}}{{x}+\mathrm{4}}=\frac{{x}+\mathrm{4}−\mathrm{4}}{{x}+\mathrm{4}}=\mathrm{1}−\frac{\mathrm{4}}{{x}+\mathrm{4}} \\ $$$$\Rightarrow\int\frac{{x}}{{x}+\mathrm{4}}{dx}=\int\mathrm{1}{dx}−\mathrm{4}\int\frac{\mathrm{1}}{{x}+\mathrm{4}}{dx} \\ $$$$={x}−\mathrm{4}{ln}\mid{x}+\mathrm{4}\mid+{c} \\…
Question Number 206645 by Red1ight last updated on 21/Apr/24 $$\mathrm{Let}\:{f}\left({x}\right)={x}\left({x}−\mathrm{10}\right) \\ $$$$\mathrm{and}\:\mathrm{let}\:\mathrm{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{region}\:\mathrm{enclosed}\:\mathrm{within} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{points} \\ $$$$\left(\mathrm{2},\mathrm{7}\right),\left(\mathrm{8},\mathrm{7}\right),\left(\mathrm{2},\mathrm{4}\right),\left(\mathrm{8},\mathrm{4}\right) \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{average}\:\mathrm{arc}\:\mathrm{length}\:\mathrm{of}\:\mathrm{a}\centerdot{f}\left({x}\right) \\ $$$$\mathrm{inside}\:\mathrm{A},{a}\in\mathbb{R}^{−} \\ $$ Commented by mr…
Question Number 206642 by universe last updated on 21/Apr/24 Answered by MathematicalUser2357 last updated on 26/Apr/24 $$\mathrm{The}\:\mathrm{calculation}\:\mathrm{was}\:\mathrm{aborted}\:\mathrm{because}\:\mathrm{it}\:\mathrm{took}\:\mathrm{too} \\ $$$$\mathrm{long}.\:\mathrm{Please}\:\mathrm{make}\:\mathrm{sure}\:\mathrm{that}\:\mathrm{your}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{incorrect},\:\mathrm{or} \\ $$$$\mathrm{try}\:\mathrm{tosimplify}\:\mathrm{your}\:\mathrm{query}.\:{Have}\:{you}\:{tried}\:{using}\:{the} \\ $$$${option}\:{to}\:{simplify}\:{expressions}??? \\ $$…
Question Number 206639 by mathlove last updated on 21/Apr/24 Commented by Frix last updated on 21/Apr/24 $$=\underset{\frac{\pi}{\mathrm{12}}} {\overset{\frac{\pi}{\mathrm{11}}} {\int}}\left(\frac{\sqrt{{s}}}{\mathrm{2}\left(\sqrt{{c}}+\sqrt{{s}}\right)}−\frac{\sqrt{{cs}}}{{c}+\sqrt{{s}}\left(\sqrt{{c}}+\mathrm{1}\right)}+\frac{\left(\sqrt{{c}}−\mathrm{1}\right){s}+\sqrt{{cs}}}{\mathrm{2}\left(\left(\sqrt{{s}}+\mathrm{1}\right){c}+\left(\sqrt{{c}}+\mathrm{1}\right){s}\right)}\right){dx} \\ $$$$\mathrm{With}\:{c}=\mathrm{cos}\:{x}\:\wedge{s}=\mathrm{sin}\:{x} \\ $$$$\mathrm{I}\:\mathrm{doubt}\:\mathrm{we}\:\mathrm{can}\:\mathrm{solve}\:\mathrm{this}. \\ $$…
Question Number 206592 by NasaSara last updated on 19/Apr/24 Answered by namphamduc last updated on 20/Apr/24 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left({x}\right)}{\mathrm{1}−{x}}{dx}=\frac{\mathrm{4}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left({x}\right)}{\mathrm{1}−{x}}{dx}−\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left({x}\right)}{\mathrm{1}−{x}}{dx} \\ $$$${x}\rightarrow{x}^{\mathrm{2}}…
Question Number 206579 by York12 last updated on 19/Apr/24 $$\mathrm{Evaluate}\::\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{\mathrm{ln}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}}{d}\left({x}\right). \\ $$ Answered by Berbere last updated on 19/Apr/24 $$=\left[{ln}\left(\mathrm{1}+{x}\right){ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right]_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 206558 by luciferit last updated on 18/Apr/24 Answered by Frix last updated on 18/Apr/24 $$\int\sqrt{\mathrm{3}+\mathrm{5}\sqrt{{x}}}{dx}\:\overset{{t}=\sqrt{\mathrm{3}+\mathrm{5}\sqrt{{x}}}} {=}\: \\ $$$$=\frac{\mathrm{4}}{\mathrm{25}}\int\left({t}^{\mathrm{4}} −\mathrm{3}{t}^{\mathrm{2}} \right){dt}=\frac{\mathrm{4}}{\mathrm{25}}\left(\frac{{t}^{\mathrm{5}} }{\mathrm{5}}−{t}^{\mathrm{3}} \right)= \\…
Question Number 206557 by luciferit last updated on 18/Apr/24 Answered by Frix last updated on 18/Apr/24 $$\mathrm{Without}\:\mathrm{substitution}: \\ $$$$\int\frac{\mathrm{3}+\mathrm{2sin}^{\mathrm{2}} \:{x}}{\mathrm{cos}^{\mathrm{2}} \:{x}}{dx}=\int\left(\frac{\mathrm{5}}{\mathrm{cos}^{\mathrm{2}} \:{x}}−\mathrm{2}\right){dx}= \\ $$$$=\mathrm{5tan}\:{x}\:−\mathrm{2}{x}\:+{C} \\…
Question Number 206549 by Shrodinger last updated on 18/Apr/24 $$\int_{\mathrm{0}} ^{+\infty} \frac{{e}^{−{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{dx} \\ $$ Commented by Frix last updated on 18/Apr/24…