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Question Number 56329    Answers: 0   Comments: 1

1)calculate A_n =∫_(1/n) ^1 ((ln(1+x^2 ))/(1+x^2 ))dx with n integr and n≥1 2) find lim_(n→+∞) A_n 3) study the convergence of Σ A_n

$$\left.\mathrm{1}\right){calculate}\:{A}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$

Question Number 56345    Answers: 0   Comments: 1

let f(a) =∫_0 ^∞ (dx/(x^n +a^n )) with n integr ≥2 and a>0 1) calculate f(a) intems of a 2) let g(a) =∫_0 ^∞ (dx/((x^n +a^n )^2 )) calculate g(a) interms of a 3) find the values of integrals ∫_0 ^∞ (dx/(x^8 +16)) and ∫_0 ^∞ (dx/((x^8 +16)^2 ))

$${let}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{{x}^{{n}} \:+{a}^{{n}} }\:\:\:{with}\:{n}\:{integr}\:\geqslant\mathrm{2}\:\:{and}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{intems}\:{of}\:{a} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{{n}} \:+{a}^{{n}} \right)^{\mathrm{2}} }\:\:{calculate}\:{g}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{8}} +\mathrm{16}}\:\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{8}} \:+\mathrm{16}\right)^{\mathrm{2}} } \\ $$

Question Number 56500    Answers: 1   Comments: 1

find the X_0 value of the x variable with which the function assumes the lower value. f(x)=x^2 −3x+7+sin(πx)

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{X}}_{\mathrm{0}} \:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{x}}\:\boldsymbol{{variable}}\:\boldsymbol{{with}}\:\boldsymbol{{which}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{function}}\:\boldsymbol{{assumes}}\:\boldsymbol{{the}}\:\boldsymbol{{lower}}\:\boldsymbol{{value}}. \\ $$$$ \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{7}+{sin}\left(\pi{x}\right) \\ $$

Question Number 56503    Answers: 1   Comments: 0

A boat travels 30km upstream and 44km downstream in 10 hours. in 13 hours it can travel 40km upstream and 55km downstream. Determine the speed of the stream and that of the boat in still water. (in km/hr)

$$\mathrm{A}\:\mathrm{boat}\:\mathrm{travels}\:\mathrm{30km}\:\mathrm{upstream}\:\mathrm{and}\: \\ $$$$\mathrm{44km}\:\mathrm{downstream}\:\mathrm{in}\:\mathrm{10}\:\mathrm{hours}.\: \\ $$$$\mathrm{in}\:\mathrm{13}\:\mathrm{hours}\:\mathrm{it}\:\mathrm{can}\:\mathrm{travel}\:\mathrm{40km}\:\mathrm{upstream} \\ $$$$\mathrm{and}\:\mathrm{55km}\:\mathrm{downstream}.\:\mathrm{Determine}\:\mathrm{the} \\ $$$$\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{stream}\:\:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{boat}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water}.\:\left(\mathrm{in}\:\mathrm{km}/\mathrm{hr}\right) \\ $$

Question Number 56325    Answers: 0   Comments: 3

Happy π−day

$${Happy}\:\pi−{day} \\ $$

Question Number 56311    Answers: 0   Comments: 1

let f(x) =∫_0 ^∞ ((cos(xt))/(x^2 +t^2 )) dt with x>0 1) find f(x) 2) find the values of ∫_0 ^∞ ((cos(t))/(1+t^2 ))dt and ∫_0 ^∞ ((cos(2t))/(4+t^2 ))dt 3) let U_n =∫_0 ^∞ ((cos(nt))/(n^2 +t^2 ))dt find lim_(n→+∞) U_n and study the convergenge of Σ U_n and Σ U_n ^2

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\mathrm{2}{t}\right)}{\mathrm{4}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({nt}\right)}{{n}^{\mathrm{2}} +{t}^{\mathrm{2}} }{dt}\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \:\:\:\:{and}\:{study}\:{the}\:{convergenge}\:{of} \\ $$$$\Sigma\:{U}_{{n}} \:\:\:{and}\:\Sigma\:{U}_{{n}} ^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 56310    Answers: 0   Comments: 2

let f(x)=∫_(−∞) ^(+∞) cos(t^2 +xt +3)dt with x>0 1) find f(x) 2) calculate ∫_1 ^4 f(x)dx and ∫_1 ^(+∞) f(x)dx

$${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:{cos}\left({t}^{\mathrm{2}} \:+{xt}\:+\mathrm{3}\right){dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{4}} {f}\left({x}\right){dx}\:{and}\:\int_{\mathrm{1}} ^{+\infty} {f}\left({x}\right){dx} \\ $$

Question Number 56321    Answers: 2   Comments: 0

Solve for x and y x (√x) + y(√y) = 182 ..... (i) x (√y) + y(√x) = 183 ..... (ii)

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\:\mathrm{x}\:\sqrt{\mathrm{x}}\:\:+\:\mathrm{y}\sqrt{\mathrm{y}}\:\:=\:\mathrm{182}\:\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\mathrm{x}\:\sqrt{\mathrm{y}}\:\:+\:\mathrm{y}\sqrt{\mathrm{x}}\:\:=\:\mathrm{183}\:\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$

Question Number 56316    Answers: 1   Comments: 1

Question Number 56301    Answers: 2   Comments: 1

Minimum value of b that satisfy the following inequality ((53)/(201)) < (a/b) < (4/(15)) is ...

$${Minimum}\:\:{value}\:\:{of}\:\:{b}\:\:{that}\:\:{satisfy}\:\:{the} \\ $$$${following}\:\:{inequality}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{53}}{\mathrm{201}}\:\:<\:\:\frac{{a}}{{b}}\:\:<\:\:\frac{\mathrm{4}}{\mathrm{15}}\:\:\:\:\:{is}\:\:\:... \\ $$

Question Number 56289    Answers: 1   Comments: 7

Question Number 56282    Answers: 0   Comments: 5

Question Number 56280    Answers: 2   Comments: 2

Evaluate : 1) ((∫_0 ^( 1_ ) (1−(1−x^2 )^(100) )^(201) .xdx)/(∫_0 ^( 1) (1−(1−x^2 )^(100) )^(202) .xdx)) = ? 2) ((∫_0 ^( 1) (1−x^(200) )^(201) dx)/(∫_0 ^( 1) (1−x^(200) )^(202) dx)) = ?

$${Evaluate}\:: \\ $$$$\left.\mathrm{1}\right)\:\frac{\int_{\mathrm{0}} ^{\:\mathrm{1}_{} } \left(\mathrm{1}−\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{100}} \right)^{\mathrm{201}} \:.{xdx}}{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{100}} \right)^{\mathrm{202}} .{xdx}}\:=\:? \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\frac{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{200}} \right)^{\mathrm{201}} {dx}}{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{200}} \right)^{\mathrm{202}} {dx}}\:=\:? \\ $$

Question Number 56264    Answers: 0   Comments: 3

(x_C −h)^2 +3((s/2)−x_C )^2 = a^2 (x_A −h)^2 +3((s/2)+x_A )^2 = c^2 (x_C −x_A )^2 = b^2 /4 .

$$\left({x}_{{C}} −{h}\right)^{\mathrm{2}} +\mathrm{3}\left(\frac{{s}}{\mathrm{2}}−{x}_{{C}} \right)^{\mathrm{2}} \:=\:{a}^{\mathrm{2}} \\ $$$$\:\left({x}_{{A}} −{h}\right)^{\mathrm{2}} +\mathrm{3}\left(\frac{{s}}{\mathrm{2}}+{x}_{{A}} \right)^{\mathrm{2}} =\:{c}^{\mathrm{2}} \\ $$$$\:\:\left({x}_{{C}} −{x}_{{A}} \right)^{\mathrm{2}} \:=\:{b}^{\mathrm{2}} /\mathrm{4}\:. \\ $$

Question Number 56245    Answers: 2   Comments: 1

Question Number 56244    Answers: 0   Comments: 7

Is ∞ a complex number. If not so what is It.

$${Is}\:\infty\:{a}\:{complex}\:{number}. \\ $$$${If}\:{not}\:{so}\:{what}\:{is}\:{It}. \\ $$

Question Number 56243    Answers: 1   Comments: 0

6/2×5 which one correct 6/2×5 6/2×5 =3×5 =6/10 =15 =0.6

$$\mathrm{6}/\mathrm{2}×\mathrm{5} \\ $$$$\:{which}\:{one}\:{correct} \\ $$$$\mathrm{6}/\mathrm{2}×\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}/\mathrm{2}×\mathrm{5} \\ $$$$=\mathrm{3}×\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{6}/\mathrm{10} \\ $$$$=\mathrm{15}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{0}.\mathrm{6} \\ $$

Question Number 56229    Answers: 2   Comments: 4

How can you prove (not geometrically) the following? Σ_(k = 0) ^n k = (( n ( n + 1 ) )/2)

$$\mathrm{How}\:\mathrm{can}\:\mathrm{you}\:\mathrm{prove}\:\left(\mathrm{not}\:\mathrm{geometrically}\right) \\ $$$$\mathrm{the}\:\mathrm{following}? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\underset{{k}\:=\:\mathrm{0}} {\overset{{n}} {\sum}}{k}\:\:=\:\:\frac{\:{n}\:\left(\:{n}\:+\:\mathrm{1}\:\right)\:}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 56215    Answers: 4   Comments: 2

(√(x/(x−1)))+(√((x−1)/x))=2 find x

$$\sqrt{\frac{{x}}{{x}−\mathrm{1}}}+\sqrt{\frac{{x}−\mathrm{1}}{{x}}}=\mathrm{2} \\ $$$$ \\ $$$${find}\:{x} \\ $$

Question Number 56214    Answers: 3   Comments: 0

x^x =4 find x

$${x}^{{x}} =\mathrm{4} \\ $$$$ \\ $$$${find}\:{x} \\ $$

Question Number 56213    Answers: 1   Comments: 1

xsin x=5 find x

$${x}\mathrm{sin}\:{x}=\mathrm{5} \\ $$$$ \\ $$$${find}\:{x} \\ $$

Question Number 56212    Answers: 1   Comments: 0

In how many ways can 4 boys and 3 girls stand in a straight line a. if there are no restrictions b. if the boys stand next to each other

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{4}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{3}\:\mathrm{girls} \\ $$$$\mathrm{stand}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line} \\ $$$$\mathrm{a}.\:\mathrm{if}\:\mathrm{there}\:\mathrm{are}\:\mathrm{no}\:\mathrm{restrictions} \\ $$$$\mathrm{b}.\:\mathrm{if}\:\mathrm{the}\:\mathrm{boys}\:\mathrm{stand}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other} \\ $$

Question Number 56205    Answers: 1   Comments: 0

find (or prove it can′t exist) a f:R→R diferentiable such that ∫_(a−δ) ^(a+δ) f(x)dx=0,∀a∈R,δ>0 (df/dx)=0,∀x∈R

$$\mathrm{find}\:\left(\mathrm{or}\:\mathrm{prove}\:\mathrm{it}\:\mathrm{can}'\mathrm{t}\:\mathrm{exist}\right)\:\mathrm{a}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{diferentiable} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{a}−\delta} {\overset{{a}+\delta} {\int}}{f}\left({x}\right){dx}=\mathrm{0},\forall{a}\in\mathbb{R},\delta>\mathrm{0} \\ $$$$\frac{{df}}{{dx}}=\mathrm{0},\forall{x}\in\mathbb{R} \\ $$

Question Number 56203    Answers: 0   Comments: 0

Question Number 56202    Answers: 1   Comments: 0

lim_(x→0) ((x^2 tan^(−1) (x) − 3 ∫_0 ^x sin (t^2 ) dt)/x^5 ) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\:\frac{{x}^{\mathrm{2}} \:\mathrm{tan}^{−\mathrm{1}} \left({x}\right)\:−\:\mathrm{3}\:\underset{\mathrm{0}} {\int}\:\overset{{x}} {\:}\:\mathrm{sin}\:\left({t}^{\mathrm{2}} \right)\:{dt}}{{x}^{\mathrm{5}} }\:\:=\:\:? \\ $$

Question Number 56200    Answers: 0   Comments: 5

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