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Question Number 109169 Answers: 1 Comments: 0

The value of 1∙1! + 2∙2! + 3∙3! +...+n∙n! is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{1}\centerdot\mathrm{1}!\:+\:\mathrm{2}\centerdot\mathrm{2}!\:+\:\mathrm{3}\centerdot\mathrm{3}!\:+...+{n}\centerdot{n}! \\ $$$$\mathrm{is}\: \\ $$

Question Number 109167 Answers: 1 Comments: 0

Question Number 109160 Answers: 4 Comments: 0

Find the equation of line through the point of intersection of the line x+3y−11=0 and 5x−4y+2=0 and perpendicular to 4x+2y+9=0.

$${Find}\:{the}\:{equation}\:{of}\:{line}\:{through} \\ $$$${the}\:{point}\:{of}\:{intersection}\:{of}\:{the} \\ $$$${line}\:{x}+\mathrm{3}{y}−\mathrm{11}=\mathrm{0}\:{and}\:\mathrm{5}{x}−\mathrm{4}{y}+\mathrm{2}=\mathrm{0} \\ $$$${and}\:{perpendicular}\:{to}\:\mathrm{4}{x}+\mathrm{2}{y}+\mathrm{9}=\mathrm{0}. \\ $$

Question Number 109149 Answers: 0 Comments: 2

if f:x→x be a mapping prove that (f⊆I_X ∨I_X ⊆f)→f=I_(X ) ? help me sir

$${if}\:{f}:{x}\rightarrow{x}\:{be}\:{a}\:{mapping}\:{prove}\:{that}\:\left({f}\subseteq{I}_{{X}} \vee{I}_{{X}} \subseteq{f}\right)\rightarrow{f}={I}_{{X}\:} ? \\ $$$${help}\:{me}\:{sir} \\ $$

Question Number 109147 Answers: 3 Comments: 0

The principal argument of z=1+cos (((6π)/5))+isin (((6π)/5)) is = ?

$${The}\:{principal}\:{argument}\:{of} \\ $$$${z}=\mathrm{1}+\mathrm{cos}\:\left(\frac{\mathrm{6}\pi}{\mathrm{5}}\right)+{i}\mathrm{sin}\:\left(\frac{\mathrm{6}\pi}{\mathrm{5}}\right)\:\:\:{is}\:=\:? \\ $$

Question Number 109145 Answers: 0 Comments: 0

Question Number 109144 Answers: 0 Comments: 0

Question Number 109143 Answers: 0 Comments: 0

Question Number 109142 Answers: 0 Comments: 1

Question Number 109141 Answers: 2 Comments: 0

Question Number 109139 Answers: 0 Comments: 1

lim_(x→−∞) (1−x) sin ∣2x−2∣ ?

$$\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\left(\mathrm{1}−{x}\right)\:\mathrm{sin}\:\mid\mathrm{2}{x}−\mathrm{2}\mid\:? \\ $$

Question Number 109136 Answers: 1 Comments: 0

∫_0 ^(1/2) ((ln(1-t)ln(t))/t) dt I′m about to give up

$$\int_{\mathrm{0}} ^{\mathrm{1}/\mathrm{2}} \frac{{ln}\left(\mathrm{1}-{t}\right){ln}\left({t}\right)}{{t}}\:{dt} \\ $$$${I}'{m}\:{about}\:{to}\:{give}\:{up} \\ $$

Question Number 109237 Answers: 4 Comments: 0

((♭o♭hans)/(∠∠∠∠∠)) { ((x^3 +x^2 y = 9)),((y^3 +y^2 x = 25)) :}. find x and y.

$$\:\:\frac{\flat{o}\flat{hans}}{\angle\angle\angle\angle\angle} \\ $$$$\begin{cases}{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} {y}\:=\:\mathrm{9}}\\{{y}^{\mathrm{3}} +{y}^{\mathrm{2}} {x}\:=\:\mathrm{25}}\end{cases}.\:{find}\:{x}\:{and}\:{y}. \\ $$

Question Number 109129 Answers: 1 Comments: 0

∫_0 ^(π/2) ((ln (cos x)ln (sin x))/(tan x)) dx

$$\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)\mathrm{ln}\:\left(\mathrm{sin}\:{x}\right)}{\mathrm{tan}\:{x}}\:{dx} \\ $$

Question Number 109128 Answers: 0 Comments: 1

A high school question from Japan Let P(x) be a real polynomial of degree 4 and P^((4)) (0)=72. If there exists t∈R and m,n∈R such that P′′(t)=0 and P′(t)=P′(t+m)=∫_t ^( t+m) P′(x)dx=n ⇒Find the value of m and n.

$$\mathrm{A}\:\mathrm{high}\:\mathrm{school}\:\mathrm{question}\:\mathrm{from}\:\mathrm{Japan} \\ $$$$ \\ $$$$\mathrm{Let}\:\mathrm{P}\left({x}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{polynomial}\:\mathrm{of}\:\mathrm{degree}\:\mathrm{4}\: \\ $$$$\mathrm{and}\:\mathrm{P}^{\left(\mathrm{4}\right)} \left(\mathrm{0}\right)=\mathrm{72}. \\ $$$$\mathrm{If}\:\mathrm{there}\:\mathrm{exists}\:{t}\in\mathbb{R}\:\mathrm{and}\:{m},{n}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{P}''\left({t}\right)=\mathrm{0}\:\mathrm{and}\:\mathrm{P}'\left({t}\right)=\mathrm{P}'\left({t}+{m}\right)=\int_{{t}} ^{\:{t}+{m}} \mathrm{P}'\left({x}\right){dx}={n} \\ $$$$\Rightarrow\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}\:\mathrm{and}\:{n}. \\ $$

Question Number 109123 Answers: 0 Comments: 0

prove that : ∫_(π/4) ^((3π)/4) sin(x)−cos(x)dx ≥∫_π ^((3π)/2) sin(x)+cos(x)dx

$${prove}\:{that}\:: \\ $$$$\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\mathrm{3}\pi}{\mathrm{4}}} {sin}\left({x}\right)−{cos}\left({x}\right){dx}\:\geqslant\int_{\pi} ^{\frac{\mathrm{3}\pi}{\mathrm{2}}} {sin}\left({x}\right)+{cos}\left({x}\right){dx} \\ $$

Question Number 109119 Answers: 2 Comments: 0

if f(x^2 )=y ,f′(x)=(√(5x−1 )) then (dy/dx)=.....

$${if}\:{f}\left({x}^{\mathrm{2}} \right)={y}\:\:,{f}'\left({x}\right)=\sqrt{\mathrm{5}{x}−\mathrm{1}\:}\:{then}\: \\ $$$$\frac{{dy}}{{dx}}=..... \\ $$

Question Number 109117 Answers: 1 Comments: 0

Question Number 109116 Answers: 3 Comments: 0

∫_0 ^∞ ((ln(x))/((1+x)^4 )) Please help

$$\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{4}} } \\ $$$${Please}\:{help} \\ $$

Question Number 109101 Answers: 3 Comments: 0

Given a function f(x+3)=f(x) for ∀x∈R. If ∫_(−3) ^6 f(x)dx = −6 then ∫_3 ^9 f(x) dx = ?

$$\:{Given}\:{a}\:{function}\:{f}\left({x}+\mathrm{3}\right)={f}\left({x}\right) \\ $$$${for}\:\forall{x}\in\mathbb{R}.\:{If}\:\underset{−\mathrm{3}} {\overset{\mathrm{6}} {\int}}{f}\left({x}\right){dx}\:=\:−\mathrm{6}\: \\ $$$${then}\:\underset{\mathrm{3}} {\overset{\mathrm{9}} {\int}}{f}\left({x}\right)\:{dx}\:=\:? \\ $$

Question Number 109097 Answers: 1 Comments: 0

((♭o♭hans)/(∼∼∼∼∼)) ∫_1 ^2 x sec^(−1) (x)dx=?

$$\:\:\frac{\boldsymbol{\flat{o}\flat{hans}}}{\sim\sim\sim\sim\sim} \\ $$$$\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}{x}\:\mathrm{sec}^{−\mathrm{1}} \left({x}\right){dx}=? \\ $$

Question Number 109091 Answers: 2 Comments: 0

((△♭eMath▽)/(≡⊸≡⊸≡)) lim_(x→π) ((sin x)/( (√(π+tan x))−(√(π−tan x)))) ?

$$\:\:\:\:\frac{\bigtriangleup\flat{e}\mathscr{M}{ath}\bigtriangledown}{\equiv\multimap\equiv\multimap\equiv} \\ $$$$\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}}{\:\sqrt{\pi+\mathrm{tan}\:{x}}−\sqrt{\pi−\mathrm{tan}\:{x}}}\:? \\ $$

Question Number 109090 Answers: 1 Comments: 0

Find all those roots of the equation z^(12) −56z^6 −512=0 whose imaginary part is positive.

$${Find}\:{all}\:{those}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\boldsymbol{{z}}^{\mathrm{12}} −\mathrm{56}\boldsymbol{{z}}^{\mathrm{6}} −\mathrm{512}=\mathrm{0}\:\:{whose}\:{imaginary} \\ $$$${part}\:{is}\:{positive}. \\ $$

Question Number 109086 Answers: 1 Comments: 0

♭_→ o_→ ♭h_⊸ ans_⊸ (1) (x^2 e^(−(y/x)) +y^2 ) dx = xy dy (2)(((f(x))/x))′ = x^2 e^(−x^2 ) ; f(1) = (1/e) g(x) = (4/e^4 )∫_1 ^x e^t^2 f(t) dt . find f(2)−g(2)

$$\:\:\:\underset{\rightarrow} {\flat}\underset{\rightarrow} {{o}}\flat\underset{\multimap} {{h}an}\underset{\multimap} {{s}} \\ $$$$\left(\mathrm{1}\right)\:\left({x}^{\mathrm{2}} {e}^{−\frac{{y}}{{x}}} +{y}^{\mathrm{2}} \right)\:{dx}\:=\:{xy}\:{dy}\: \\ $$$$\left(\mathrm{2}\right)\left(\frac{{f}\left({x}\right)}{{x}}\right)'\:=\:{x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } \:;\:{f}\left(\mathrm{1}\right)\:=\:\frac{\mathrm{1}}{{e}}\: \\ $$$$\:\:\:\:\:\:\:{g}\left({x}\right)\:=\:\frac{\mathrm{4}}{{e}^{\mathrm{4}} }\underset{\mathrm{1}} {\overset{{x}} {\int}}{e}^{{t}^{\mathrm{2}} } \:{f}\left({t}\right)\:{dt}\:.\:{find}\:{f}\left(\mathrm{2}\right)−{g}\left(\mathrm{2}\right) \\ $$

Question Number 109082 Answers: 1 Comments: 0

prove that : ∫_(−(π/2)) ^(−(π/4)) 2cos(x)+sin(x)dx≤∫_(−(π/2)) ^(−(π/4)) cos(x)−sin(x)dx

$${prove}\:{that}\:: \\ $$$$\int_{−\frac{\pi}{\mathrm{2}}} ^{−\frac{\pi}{\mathrm{4}}} \mathrm{2}{cos}\left({x}\right)+{sin}\left({x}\right){dx}\leqslant\int_{−\frac{\pi}{\mathrm{2}}} ^{−\frac{\pi}{\mathrm{4}}} {cos}\left({x}\right)−{sin}\left({x}\right){dx} \\ $$

Question Number 109075 Answers: 1 Comments: 0

Given the equations of twe circles C_1 : x^2 + y^2 −6x−4y + 9 = 0 and C_2 : x^2 + y^2 −2x−6y + 9. (a) Find the equation of the circle C_3 which passes through the centre of C_1 and through the point of intersection of C_1 and C_2 . (b) The equations of two tangents from the origin to C_1 and the lenght of each tangent.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{twe}\:\mathrm{circles}\: \\ $$$${C}_{\mathrm{1}} \::\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:−\mathrm{6}{x}−\mathrm{4}{y}\:+\:\mathrm{9}\:=\:\mathrm{0}\:\mathrm{and}\:{C}_{\mathrm{2}} \::\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{6}{y}\:+\:\mathrm{9}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:{C}_{\mathrm{3}} \:\mathrm{which}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre} \\ $$$$\mathrm{of}\:{C}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{of}\:{C}_{\mathrm{1}} \:\mathrm{and}\:{C}_{\mathrm{2}} . \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{two}\:\mathrm{tangents}\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{to}\:{C}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{lenght} \\ $$$$\mathrm{of}\:\mathrm{each}\:\mathrm{tangent}. \\ $$

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