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Question Number 68022    Answers: 1   Comments: 1

Question Number 68021    Answers: 1   Comments: 1

Question Number 68019    Answers: 0   Comments: 5

let F(x) =∫_x ^(x^2 +1) e^(−2t) sin(xt)dt determine F^′ (x) and calculate lim_(x→0) F(x).

$${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} +\mathrm{1}} {e}^{−\mathrm{2}{t}} {sin}\left({xt}\right){dt} \\ $$$${determine}\:{F}\:^{'} \left({x}\right)\:{and}\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}\left({x}\right). \\ $$$$ \\ $$

Question Number 68001    Answers: 0   Comments: 1

let F(x)=∫_(2x) ^(x^2 +1) (e^(−xt) /(x+2t))dt calculate F^′ (x)

$${let}\:{F}\left({x}\right)=\int_{\mathrm{2}{x}} ^{{x}^{\mathrm{2}} +\mathrm{1}} \:\:\frac{{e}^{−{xt}} }{{x}+\mathrm{2}{t}}{dt}\:\:\:\:{calculate}\:{F}\:^{'} \left({x}\right) \\ $$

Question Number 67997    Answers: 0   Comments: 0

5y^2 +2axy+b=0 ay^2 +2bx+5c=0 (5x+3a)y^2 +(4ax^2 )y−bx−5c=0 5y^2 −x(5x+2a)y−ax^3 −3b=0 Please solve simultaneously for x and y such that all four equations are obeyed.

$$\mathrm{5}{y}^{\mathrm{2}} +\mathrm{2}{axy}+{b}=\mathrm{0} \\ $$$${ay}^{\mathrm{2}} +\mathrm{2}{bx}+\mathrm{5}{c}=\mathrm{0} \\ $$$$\left(\mathrm{5}{x}+\mathrm{3}{a}\right){y}^{\mathrm{2}} +\left(\mathrm{4}{ax}^{\mathrm{2}} \right){y}−{bx}−\mathrm{5}{c}=\mathrm{0} \\ $$$$\mathrm{5}{y}^{\mathrm{2}} −{x}\left(\mathrm{5}{x}+\mathrm{2}{a}\right){y}−{ax}^{\mathrm{3}} −\mathrm{3}{b}=\mathrm{0} \\ $$$${Please}\:{solve}\:{simultaneously} \\ $$$${for}\:{x}\:{and}\:{y}\:{such}\:{that}\:{all}\:{four} \\ $$$${equations}\:{are}\:{obeyed}. \\ $$

Question Number 67996    Answers: 1   Comments: 4

Question Number 67992    Answers: 1   Comments: 0

(1) z=a+bi (2) z=re^(iθ) express the values of (a) real (z^z ) [real part] (b) imag (z^z ) [imaginary part] (c) abs (z^z ) [absolute value] (d) arg (z^z ) [argument = angle]

$$\left(\mathrm{1}\right)\:{z}={a}+{b}\mathrm{i} \\ $$$$\left(\mathrm{2}\right)\:{z}={r}\mathrm{e}^{\mathrm{i}\theta} \\ $$$$\mathrm{express}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{real}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{real}\:\mathrm{part}\right] \\ $$$$\left(\mathrm{b}\right)\:\mathrm{imag}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{imaginary}\:\mathrm{part}\right] \\ $$$$\left(\mathrm{c}\right)\:\mathrm{abs}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{absolute}\:\mathrm{value}\right] \\ $$$$\left(\mathrm{d}\right)\:\mathrm{arg}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{argument}\:=\:\mathrm{angle}\right] \\ $$

Question Number 67991    Answers: 1   Comments: 1

Question Number 67983    Answers: 2   Comments: 1

Question Number 67977    Answers: 1   Comments: 1

Question Number 67974    Answers: 0   Comments: 2

let F(x) =∫_x ^x^2 ((arctan(xt))/(x^2 +t^2 ))dt calculate F^′ (x).

$${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{arctan}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }{dt}\:\:{calculate}\:{F}\:^{'} \left({x}\right). \\ $$

Question Number 67973    Answers: 0   Comments: 5

Question Number 67972    Answers: 2   Comments: 0

if F(x)=∫_(u(x)) ^(v(x)) g(x,t)dt determine a expression for F^′ (x).

$${if}\:{F}\left({x}\right)=\int_{{u}\left({x}\right)} ^{{v}\left({x}\right)} {g}\left({x},{t}\right){dt}\:\:\:\:\:{determine}\:{a}\:{expression}\:{for}\:{F}\:^{'} \left({x}\right). \\ $$

Question Number 67969    Answers: 0   Comments: 0

Two triangles △_1 and △_2 are given,such that length of sides of triangle 1,are equail to length of medians of triangle 2. 1.find the ratio of areas of triangles. 2.given that small side of △_1 , be equail to:(√2) and one angle be:90^• . find at least one angle of △_2 . 3.solve part#2,if replace: △_2 with: △_1 . 4.solve part#2,if great side of:△_1 ,be equail to :(√2).

$$\boldsymbol{\mathrm{Two}}\:\boldsymbol{\mathrm{triangles}}\:\bigtriangleup_{\mathrm{1}} \:\boldsymbol{\mathrm{and}}\:\bigtriangleup_{\mathrm{2}} \:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{given}},\boldsymbol{\mathrm{such}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{length}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{sides}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{triangle}}\:\mathrm{1},\boldsymbol{\mathrm{are}}\: \\ $$$$\boldsymbol{\mathrm{equail}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{length}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{medians}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{triangle}}\:\mathrm{2}. \\ $$$$\mathrm{1}.\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{ratio}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{areas}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{triangles}}. \\ $$$$\mathrm{2}.\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{small}}\:\boldsymbol{\mathrm{side}}\:\boldsymbol{\mathrm{of}}\:\bigtriangleup_{\mathrm{1}} ,\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{equail}}\:\boldsymbol{\mathrm{to}}:\sqrt{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{one}}\:\boldsymbol{\mathrm{angle}}\:\boldsymbol{\mathrm{be}}:\mathrm{90}^{\bullet} . \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{least}}\:\boldsymbol{\mathrm{one}}\:\boldsymbol{\mathrm{angle}}\:\boldsymbol{\mathrm{of}}\:\bigtriangleup_{\mathrm{2}} . \\ $$$$\mathrm{3}.\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{part}}#\mathrm{2},\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{replace}}:\:\bigtriangleup_{\mathrm{2}} \boldsymbol{\mathrm{with}}:\:\bigtriangleup_{\mathrm{1}} . \\ $$$$\mathrm{4}.\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{part}}#\mathrm{2},\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{great}}\:\boldsymbol{\mathrm{side}}\:\boldsymbol{\mathrm{of}}:\bigtriangleup_{\mathrm{1}} ,\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{equail}}\: \\ $$$$\boldsymbol{\mathrm{to}}\::\sqrt{\mathrm{2}}. \\ $$

Question Number 67963    Answers: 1   Comments: 1

Question Number 67960    Answers: 0   Comments: 2

(d/dx)[tan^(−1) ((4x)/(√(1−4x^2 )))] or (d/dx)tan^(−1) (2tanθ) [where 2x=sinθ ] which comes later if done considering 2x=sinθ please help

$$\frac{{d}}{{dx}}\left[{tan}^{−\mathrm{1}} \frac{\mathrm{4}{x}}{\sqrt{\mathrm{1}−\mathrm{4}{x}^{\mathrm{2}} }}\right] \\ $$$${or} \\ $$$$\frac{{d}}{{dx}}{tan}^{−\mathrm{1}} \left(\mathrm{2}{tan}\theta\right)\:\:\:\:\:\:\:\left[{where}\:\mathrm{2}{x}={sin}\theta\:\right] \\ $$$$\:\:\:{which}\:{comes}\:{later}\:{if}\:{done}\:{considering} \\ $$$$\mathrm{2}{x}={sin}\theta \\ $$$${please}\:{help} \\ $$

Question Number 67959    Answers: 0   Comments: 1

∫(√(e^y^2 )) dy pleas sir can you help me?

$$\int\sqrt{{e}^{{y}^{\mathrm{2}} } \:\:}\:{dy}\:\:{pleas}\:{sir}\:{can}\:{you}\:{help}\:{me}? \\ $$

Question Number 67958    Answers: 0   Comments: 0

lim_(x→2) (((7^((log_x (256)))^(1/3) −49)/(2^(−(√2^x )) −(1/4)))) ≈ ?

$$\: \\ $$$$\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{2}} {\boldsymbol{\mathrm{lim}}}\left(\frac{\mathrm{7}^{\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{x}}} \left(\mathrm{256}\right)}} −\mathrm{49}}{\mathrm{2}^{−\sqrt{\mathrm{2}^{\boldsymbol{\mathrm{x}}} }} −\frac{\mathrm{1}}{\mathrm{4}}}\right)\:\approx\:? \\ $$$$\: \\ $$

Question Number 67948    Answers: 0   Comments: 0

Question Number 67946    Answers: 0   Comments: 3

use Green−Riemann formuler to determined: I=∫∫_D xydxdy D={(x,y)∈R^2 ∣x≥0;y≥;x+y≤1}

$$\mathrm{use}\:\boldsymbol{\mathrm{Green}}−\boldsymbol{\mathrm{Riemann}}\:\boldsymbol{\mathrm{formuler}} \\ $$$$\mathrm{to}\:\mathrm{determined}: \\ $$$$\boldsymbol{\mathrm{I}}=\int\int_{\boldsymbol{\mathrm{D}}} \boldsymbol{\mathrm{xy}}\mathrm{dxdy} \\ $$$$\boldsymbol{\mathrm{D}}=\left\{\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{R}^{\mathrm{2}} \mid\mathrm{x}\geqslant\mathrm{0};\mathrm{y}\geqslant;\mathrm{x}+{y}\leqslant\mathrm{1}\right\} \\ $$

Question Number 67943    Answers: 0   Comments: 0

Question Number 67942    Answers: 0   Comments: 1

∫e^(y^2 /2) dy

$$\int{e}^{{y}^{\mathrm{2}} /\mathrm{2}} \:\:{dy} \\ $$

Question Number 67939    Answers: 1   Comments: 2

Question Number 67937    Answers: 0   Comments: 1

Question Number 67932    Answers: 1   Comments: 4

let A(θ) = ∫_0 ^∞ (dx/((x^2 +3)(x^4 −e^(iθ) ))) with 0<θ<(π/2) 1) calculate A(θ) interms of θ 2) determine also ∫_0 ^∞ (dx/((x^2 +3)(x^4 −e^(iθ) )^2 ))

$${let}\:{A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta} \right)}\:\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}\left(\theta\right)\:{interms}\:{of}\:\theta \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)\left({x}^{\mathrm{4}} −{e}^{{i}\theta} \right)^{\mathrm{2}} } \\ $$

Question Number 67931    Answers: 0   Comments: 0

let A_n =∫_0 ^(π/4) x^n {1+cosx +cos(2x)}^2 dx find a relation of recurrence betwedn the A_n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{x}^{{n}} \left\{\mathrm{1}+{cosx}\:+{cos}\left(\mathrm{2}{x}\right)\right\}^{\mathrm{2}} {dx} \\ $$$${find}\:{a}\:{relation}\:{of}\:{recurrence}\:{betwedn}\:{the}\:{A}_{{n}} \\ $$

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