Question Number 198156 by Erico last updated on 12/Oct/23 $$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{2t}−\mathrm{1}}{\mathrm{lnt}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)}=\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{t}^{\mathrm{x}} \left(\mathrm{1}−\mathrm{t}\right)^{\mathrm{1}−\mathrm{x}} \mathrm{dx} \\ $$$$\mathrm{and}\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{2t}−\mathrm{1}}{\mathrm{lnt}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)}\mathrm{dt}\:\:=\:\:\frac{\pi}{\mathrm{2}}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)}{\mathrm{sin}\left(\pi\mathrm{x}\right)}\mathrm{dx} \\ $$ Answered…
Question Number 195203 by Erico last updated on 27/Jul/23 $$\mathrm{Calculer}\:\underset{\:\mathrm{0}} {\int}^{\:+\infty} \frac{\mathrm{dt}}{\left(\mathrm{e}^{\mathrm{t}} −\mathrm{e}^{−\mathrm{t}} \right)^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} } \\ $$ Answered by Frix last updated on 27/Jul/23…
Question Number 195185 by Erico last updated on 26/Jul/23 $$\mathrm{1}/\:\:\mathrm{Montrer}\:\mathrm{que}\:\underset{\:\mathrm{0}} {\int}^{\:+\infty} \:\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}{p}−\mathrm{1}} }{\mathrm{1}−{x}^{\mathrm{4}{p}} }{dx}=\left(\frac{\mathrm{2}^{\mathrm{2}{p}−\mathrm{3}} }{{p}}\right)\pi\left[\mathrm{1}+\mathrm{2}\underset{{k}=\mathrm{1}} {\overset{{p}−\mathrm{1}} {\sum}}{cos}^{\mathrm{2}{p}−\mathrm{1}} \left(\frac{{k}\pi}{\mathrm{2}{p}}\right)\right] \\ $$$$\mathrm{2}/\:\:\:\:\mathrm{En}\:\mathrm{d}\acute {\mathrm{e}duire}\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}{p}−\mathrm{1}}…
Question Number 194915 by York12 last updated on 19/Jul/23 $$ \\ $$$$\frac{\mathrm{3}}{{x}−\mathrm{3}}+\frac{\mathrm{5}}{{x}−\mathrm{5}}+\frac{\mathrm{7}}{{x}−\mathrm{17}}+\frac{\mathrm{19}}{{x}−\mathrm{19}}={x}^{\mathrm{2}} −\mathrm{11}{x}−\mathrm{4} \\ $$ Commented by BaliramKumar last updated on 19/Jul/23 $$\mathrm{typo}\:\:\:\mathrm{7}\rightarrow\mathrm{17} \\ $$…
Question Number 194642 by horsebrand11 last updated on 12/Jul/23 $$\:\mathrm{If}\:\mathrm{A}=\begin{pmatrix}{\mathrm{a}\:\:\:\:\mathrm{b}\:\:\:\:\:\:\mathrm{c}}\\{\mathrm{b}\:\:\:\:\mathrm{c}\:\:\:\:\:\:\mathrm{a}}\\{\mathrm{c}\:\:\:\:\:\mathrm{a}\:\:\:\:\:\:\mathrm{b}}\end{pmatrix}\:\mathrm{and}\:\mathrm{a},\mathrm{b},\mathrm{c}\:>\mathrm{0} \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{abc}=\mathrm{1}\:\mathrm{and}\:\mathrm{A}^{\mathrm{T}} .\mathrm{A}=\mathrm{I} \\ $$$$\:\mathrm{find}\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} −\mathrm{3abc}\:. \\ $$ Answered by som(math1967) last updated…
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Question Number 64406 by rajesh4661kumar@gamil.com last updated on 17/Jul/19 Commented by Tony Lin last updated on 17/Jul/19 $${area}\:{of}\:\bigtriangleup=\frac{\mathrm{1}}{\mathrm{2}}\mid\begin{vmatrix}{{x}_{\mathrm{2}} −{x}_{\mathrm{1}} \:\:{y}_{\mathrm{2}} −{y}_{\mathrm{1}} }\\{{x}_{\mathrm{3}} −{x}_{\mathrm{1}} \:\:{y}_{\mathrm{3}} −{y}_{\mathrm{1}}…