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Question Number 32734    Answers: 1   Comments: 3

1) a≥0 calculate ∫_0 ^a ((n^2 −x^2 )/((n^2 +x^2 )^2 ))dx with n integr 2) find ∫_0 ^∞ ((n^2 −x^2 )/((n^2 +x^2 )^2 ))dx 3)calculate Σ_(n=1) ^∞ ∫_0 ^∞ ((n^2 −x^2 )/((n^2 +x^2 )^2 )) dx .

$$\left.\mathrm{1}\right)\:{a}\geqslant\mathrm{0}\:\:{calculate}\:\int_{\mathrm{0}} ^{{a}} \:\frac{{n}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }{\left({n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:{with} \\ $$$${n}\:{integr} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{n}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }{\left({n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right){calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }{\left({n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 32733    Answers: 0   Comments: 0

prove that Σ_(n=0) ^∞ (1/((n!)^2 )) =(1/(2π)) ∫_0 ^(2π) e^(2cosx) dx .

$${prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left({n}!\right)^{\mathrm{2}} }\:=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:{e}^{\mathrm{2}{cosx}} {dx}\:. \\ $$

Question Number 32732    Answers: 0   Comments: 0

give ∫_0 ^1 (((lnx)^p )/(1−x)) dx at form of seriewith p≥2 .

$${give}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\left({lnx}\right)^{{p}} }{\mathrm{1}−{x}}\:{dx}\:{at}\:{form}\:{of}\:{seriewith} \\ $$$${p}\geqslant\mathrm{2}\:. \\ $$

Question Number 32731    Answers: 0   Comments: 0

1) prove that ∫_0 ^1 ((arctant)/t)dt=−∫_0 ^1 ((lnt)/(1+t^2 ))dt 2) find ∫_0 ^1 ((arctant)/t)dt at form of serie

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctant}}{{t}}{dt}=−\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{lnt}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctant}}{{t}}{dt}\:{at}\:{form}\:{of}\:{serie} \\ $$

Question Number 32729    Answers: 0   Comments: 0

find lim_(n→∞) ∫_0 ^n (cos((x/n)))^n^2 dx.

$${find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{{n}} \left({cos}\left(\frac{{x}}{{n}}\right)\right)^{{n}^{\mathrm{2}} } \:{dx}. \\ $$

Question Number 32724    Answers: 0   Comments: 0

let A_n = ∫_0 ^n (√(1+(1−(x/n))^n )) dt. find a rquivalent of A_n .

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \sqrt{\mathrm{1}+\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} }\:{dt}. \\ $$$${find}\:{a}\:{rquivalent}\:{of}\:{A}_{{n}} . \\ $$

Question Number 32721    Answers: 0   Comments: 0

let x>0 and f(x)=∫_x ^(+∞) (e^(−t) /t)dt 1)calculate f^′ (x) 2) find lim_(x→+∞) xf(x) and lim_(x→0^+ ) xf(x).

$${let}\:{x}>\mathrm{0}\:{and}\:{f}\left({x}\right)=\int_{{x}} ^{+\infty} \:\:\frac{{e}^{−{t}} }{{t}}{dt} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow+\infty} {xf}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } {xf}\left({x}\right). \\ $$

Question Number 32720    Answers: 0   Comments: 0

find ∫_(−∞) ^(+∞) (dt/(1 +(t+2i)^2 )) .

$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dt}}{\mathrm{1}\:+\left({t}+\mathrm{2}{i}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 32719    Answers: 0   Comments: 0

cslculate ∫_0 ^∞ (t −[t])e^(−3t) dt .

$${cslculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\left({t}\:−\left[{t}\right]\right){e}^{−\mathrm{3}{t}} {dt}\:. \\ $$

Question Number 32718    Answers: 0   Comments: 0

find ∫_0 ^∞ arctan(2x) (e^(−tx) /x) dc with t>0 2) calculate ∫_0 ^∞ ((arctan(2x))/x) e^(−x) dx.

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:{arctan}\left(\mathrm{2}{x}\right)\:\frac{{e}^{−{tx}} }{{x}}\:{dc}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{{x}}\:{e}^{−{x}} \:{dx}. \\ $$

Question Number 32717    Answers: 0   Comments: 0

finf ∫_0 ^(+∞) (dx/(1+x^2 +x^4 ))

$${finf}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} } \\ $$

Question Number 32716    Answers: 1   Comments: 0

find ∫_0 ^(2π) ((cos^2 x)/(1+3sin^2 x))dx .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{3}{sin}^{\mathrm{2}} {x}}{dx}\:. \\ $$

Question Number 32715    Answers: 0   Comments: 1

calculate ∫_(−∞) ^(+∞) (dt/((1+it)(1+it^2 ))) .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left(\mathrm{1}+{it}\right)\left(\mathrm{1}+{it}^{\mathrm{2}} \right)}\:\:. \\ $$

Question Number 32714    Answers: 0   Comments: 1

calculate ∫_1 ^(+∞) (dt/(t^2 (√(1+t^2 )))) .

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dt}}{{t}^{\mathrm{2}} \sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:. \\ $$

Question Number 32713    Answers: 0   Comments: 1

A iniform rod of length 2a and weight W is smoothly pivoted to a fixed point at A. A load of weight 2W is attached to the end B. The rod is kept horizontally by a string attached to the midpoint D of the rod and to a point C vertically above A. If the length of the string is 2a, find, in terms of W, the tension in the string and the magnitude of the reaction at the pivot.

$$\mathrm{A}\:\mathrm{iniform}\:\mathrm{rod}\:\mathrm{of}\:\mathrm{length}\:\mathrm{2}{a}\:\mathrm{and}\:\mathrm{weight}\:{W} \\ $$$$\mathrm{is}\:\mathrm{smoothly}\:\mathrm{pivoted}\:\mathrm{to}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{point} \\ $$$$\mathrm{at}\:{A}.\:\mathrm{A}\:\mathrm{load}\:\mathrm{of}\:\mathrm{weight}\:\mathrm{2}{W}\:\mathrm{is}\:\mathrm{attached} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{end}\:{B}.\:\mathrm{The}\:\mathrm{rod}\:\mathrm{is}\:\mathrm{kept}\:\mathrm{horizontally} \\ $$$$\mathrm{by}\:\mathrm{a}\:\mathrm{string}\:\mathrm{attached}\:\mathrm{to}\:\mathrm{the}\:\mathrm{midpoint} \\ $$$${D}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rod}\:\mathrm{and}\:\mathrm{to}\:\mathrm{a}\:\mathrm{point}\:{C}\:\mathrm{vertically} \\ $$$$\mathrm{above}\:{A}.\: \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{string}\:\mathrm{is}\:\mathrm{2}{a},\:\mathrm{find}, \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{W},\:\mathrm{the}\:\mathrm{tension}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{string}\:\mathrm{and}\:\mathrm{the}\:\mathrm{magnitude}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{reaction}\:\mathrm{at}\:\mathrm{the}\:\mathrm{pivot}. \\ $$

Question Number 32712    Answers: 0   Comments: 1

calculate ∫_0 ^(π/2) (dt/(1+a cos^2 t)) .

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{cos}^{\mathrm{2}} {t}}\:. \\ $$

Question Number 32701    Answers: 1   Comments: 3

let give f(x)= (x/(√(x+1))) 1)calculate f^(−1) (x)) 2) calculate (f^(−1) )^′ (x) .

$${let}\:{give}\:{f}\left({x}\right)=\:\frac{{x}}{\sqrt{{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{1}\left.\right){calculate}\:{f}^{−\mathrm{1}} \left({x}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\left({f}^{−\mathrm{1}} \right)^{'} \left({x}\right)\:. \\ $$

Question Number 32723    Answers: 0   Comments: 0

find lim_(n→+∞) ∫_0 ^∞ e^(−t) sin^n t dt.

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}} \:{sin}^{{n}} {t}\:{dt}. \\ $$

Question Number 32722    Answers: 0   Comments: 0

find ∫_0 ^∞ (dx/(1+x^3 )) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} }\:. \\ $$

Question Number 32702    Answers: 0   Comments: 0

Question Number 32710    Answers: 1   Comments: 0

Question Number 32709    Answers: 1   Comments: 0

Question Number 32688    Answers: 1   Comments: 0

Evaluate 1) ∫_(−1) ^0 (x^2 + x 1) dx 2) ∫_1 ^5 (x −1+ (1/x^2 ))dx

$$\mathrm{Evaluate} \\ $$$$\left.\:\mathrm{1}\right)\:\int_{−\mathrm{1}} ^{\mathrm{0}} \left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:\mathrm{1}\right)\:\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{1}} ^{\mathrm{5}} \left(\mathrm{x}\:−\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\mathrm{dx} \\ $$$$ \\ $$

Question Number 32792    Answers: 1   Comments: 2

the equation 3x− 5=15 represents a straight line. a) find one point on this line. b)find the coordinates of the points when the line cuts the x−axis and the y−axis c)find the gradient of this line.

$${the}\:{equation}\: \\ $$$$\mathrm{3}{x}−\:\mathrm{5}=\mathrm{15}\:{represents}\:{a}\:{straight}\:{line}. \\ $$$$\left.{a}\right)\:{find}\:{one}\:{point}\:{on}\:{this}\:{line}. \\ $$$$\left.{b}\right){find}\:{the}\:{coordinates}\:{of}\:{the}\:{points}\:{when} \\ $$$${the}\:{line}\:{cuts}\:{the}\:{x}−{axis}\:{and}\:{the}\:{y}−{axis} \\ $$$$\left.{c}\right){find}\:{the}\:{gradient}\:{of}\:{this}\:{line}. \\ $$$$ \\ $$

Question Number 32682    Answers: 1   Comments: 0

If x_1 and x_(2 ) are roots of the equation acos 2x+bsin x = c and 2sin x_1 sinx_2 = sin x_1 +sinx_2 . Then the value of (b/(c−a)) is ?

$$\boldsymbol{{I}}{f}\:{x}_{\mathrm{1}} \:{and}\:{x}_{\mathrm{2}\:} \:{are}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${acos}\:\mathrm{2}{x}+{bsin}\:{x}\:=\:{c}\:{and}\: \\ $$$$\mathrm{2}{sin}\:{x}_{\mathrm{1}} {sinx}_{\mathrm{2}} =\:{sin}\:{x}_{\mathrm{1}} +{sinx}_{\mathrm{2}} .\:\boldsymbol{{T}}{hen}\: \\ $$$${the}\:{value}\:{of}\:\:\frac{{b}}{{c}−{a}}\:{is}\:? \\ $$

Question Number 32681    Answers: 0   Comments: 2

Total no. of polynomials of the form x^3 +ax^2 +bx+c that are divisible by x^2 +1, where a,b,c∈1,2,3,....,10 is 1) 10 2) 15 3) 5 4) 8

$$\boldsymbol{{T}}{otal}\:{no}.\:{of}\:{polynomials}\:{of}\:{the}\:{form} \\ $$$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}\:\:{that}\:{are}\:{divisible}\:{by}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{1},\:{where}\:{a},{b},{c}\in\mathrm{1},\mathrm{2},\mathrm{3},....,\mathrm{10}\:{is}\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{10} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{15} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{5} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{8} \\ $$

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