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Question Number 42559 Answers: 5 Comments: 0
Question Number 42553 Answers: 0 Comments: 1
Question Number 42550 Answers: 1 Comments: 3
$$\mathrm{2}^{\frac{\mathrm{1}}{{x}}} =\sqrt{{x}} \\ $$$${Find}\:{x} \\ $$
Question Number 43535 Answers: 2 Comments: 2
$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\sqrt{\mathrm{1}+{tan}\theta}{d}\theta\:. \\ $$
Question Number 42549 Answers: 1 Comments: 0
$$\mathrm{If}\:\:\:\mathrm{y}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{x}}} \:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{y}}} \:\:.\:\:\mathrm{show}\:\mathrm{that}\:\:\:\:\:\mathrm{x}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{z}}} \\ $$
Question Number 42543 Answers: 0 Comments: 3
Question Number 42538 Answers: 1 Comments: 1
Question Number 43537 Answers: 0 Comments: 1
$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{e}^{\frac{{k}}{{n}}} }{{n}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {S}_{{n}} \\ $$
Question Number 42521 Answers: 1 Comments: 0
$$\mathrm{Let}\:\overset{\rightarrow\:} {\mathrm{a}},\:\overset{\rightarrow} {\mathrm{b}}\:,\:\overset{\rightarrow} {\mathrm{c}}\:\mathrm{be}\:\mathrm{three}\:\mathrm{unit}\:\mathrm{vectors} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{3}\overset{\rightarrow} {\mathrm{a}}+\mathrm{4}\overset{\rightarrow} {\mathrm{b}}+\mathrm{5}\overset{\rightarrow} {\mathrm{c}}\:=\:\mathrm{0}.\:\mathrm{Then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\overset{\rightarrow\:} {\mathrm{a}},\:\overset{\rightarrow} {\mathrm{b}},\overset{\rightarrow} {\mathrm{c}}\:\mathrm{are}\:\mathrm{coplanar}. \\ $$
Question Number 42520 Answers: 1 Comments: 0
$${cos}^{\mathrm{3}} \:{A}.{sin}\mathrm{3}{A}+{sinA}.{cos}\mathrm{3}{A}=\frac{\mathrm{3}}{\mathrm{4}}{sin}\mathrm{4}{A} \\ $$
Question Number 42519 Answers: 1 Comments: 0
$${cosec}\mathrm{2}{A}+{cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}={cotA}−{cot}\mathrm{8}{A}\left({prlve}\:{ghis}\right) \\ $$
Question Number 42516 Answers: 1 Comments: 1
$$\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{a}\right),\left(\mathrm{x}−\frac{\mathrm{5}}{\mathrm{2}}\mathrm{a}\right)=? \\ $$$$\mathrm{Solve}\:\mathrm{Please}. \\ $$
Question Number 42514 Answers: 1 Comments: 0
$${calculate}\:\:{d}\left({x}!\right)/{dx}=\:? \\ $$
Question Number 42508 Answers: 2 Comments: 0
$${find}\:{the}\:{value}\:{of}\: \\ $$$${A}\:={cos}\left(\frac{\pi}{\mathrm{5}}\right).{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:{and}\:{B}\:={sin}\left(\frac{\pi}{\mathrm{5}}\right){sin}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right){sin}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right). \\ $$
Question Number 42507 Answers: 0 Comments: 2
$${let}\:{j}\:={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:\:\:{p}\left({x}\right)\:=\left(\mathrm{1}+{xj}\right)^{{n}} \:−\left(\mathrm{1}−{xj}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:{and}\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fration}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{p}\left({x}\right)}\:. \\ $$
Question Number 42506 Answers: 0 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({t}\right)}{\left(^{\mathrm{3}} \sqrt{{t}^{\mathrm{2}} }\right)\left(\mathrm{1}+{t}\right)}{dt}\:. \\ $$
Question Number 42505 Answers: 0 Comments: 2
$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{{t}}}\:{dt}\:. \\ $$
Question Number 42504 Answers: 0 Comments: 1
$${let}\:{x}>\mathrm{0}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{t}^{\mathrm{2}} } {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }\:{dt}\:=\pi\:\int_{\mathrm{0}} ^{\sqrt{{x}}} \:\:{e}^{\frac{\mathrm{1}}{{u}^{\mathrm{2}} }} \:\:{du}\:. \\ $$
Question Number 42503 Answers: 1 Comments: 1
$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} } \\ $$
Question Number 42502 Answers: 0 Comments: 0
$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\left({nx}\right)}{{cosx}\:+{sinx}}{dx}\:\:{with}\:{n}\:{from}\:{N} \\ $$
Question Number 42501 Answers: 0 Comments: 1
$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{4}{x}\right)}{{cosx}\:+{sinx}}\:\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{4}{x}\right)}{{cosx}\:+{sinx}}{dx} \\ $$
Question Number 42500 Answers: 0 Comments: 2
$${calculate}\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{{cosx}\:+{sinx}}{dx}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{cosx}\:+{sinx}}{dx} \\ $$
Question Number 42497 Answers: 0 Comments: 1
$${A}\:{body}\:{cools}\:{from}\:\mathrm{90}°{C}\:{to}\:\mathrm{40}°{C}\:{in} \\ $$$$\mathrm{2}\:{minutes}\:{at}\:{a}\:{temperature},\mathrm{20}°{C} \\ $$$${of}\:{the}\:{surrounding}.{Calculate}\:{the} \\ $$$${temperature}\:{of}\:{the}\:{body}\:{after} \\ $$$${another}\:\mathrm{5}\:{minutes}. \\ $$
Question Number 42495 Answers: 0 Comments: 0
$$\mathrm{In}\:\mathrm{the}\:\mathrm{sequence}\:\:\mathrm{1},\:\mathrm{22},\:\mathrm{333},\:...\:\mathrm{10101010101010101010},\:\mathrm{1111111111111111111111},\:... \\ $$$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{the}\:\mathrm{200th}\:\mathrm{term}\:\mathrm{is}\:?? \\ $$
Question Number 42494 Answers: 1 Comments: 1
$$\mathrm{In}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{numbers}\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{11},\:\mathrm{22},\:\mathrm{111},\:\mathrm{222},\:...\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{in}\:\mathrm{999th}\:\mathrm{terms}\:\mathrm{is}\:?? \\ $$
Question Number 42493 Answers: 0 Comments: 1
$$\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\:\:\frac{\mathrm{1}}{{i}^{{x}} {j}^{{x}} }\:\:\:{with}\:\:{x}>\mathrm{1}\:\:{for}\:{that}\:{consider} \\ $$$$\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{{x}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\:\:\:\frac{\mathrm{1}}{\left({ij}\right)^{\mathrm{2}} }\:. \\ $$
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