Question and Answers Forum

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.

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.

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

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