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Question Number 44473 Answers: 0 Comments: 1

let A_n =∫_0 ^∞ sin(n[t])e^(−t) dt 2)calculate A_n and lim_(n→+∞) n A_n 3)study the convergence of Σ_n A_n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{sin}\left({n}\left[{t}\right]\right){e}^{−{t}} {dt} \\ $$$$\left.\mathrm{2}\right){calculate}\:{A}_{{n}} \:\:{and}\:{lim}_{{n}\rightarrow+\infty} {n}\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{convergence}\:{of}\:\sum_{{n}} \:{A}_{{n}} \\ $$

Question Number 44472 Answers: 0 Comments: 1

find f(x)=∫_0 ^∞ ((ln(t)dt)/((1+xt)^2 )) withx>0

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({t}\right){dt}}{\left(\mathrm{1}+{xt}\right)^{\mathrm{2}} }\:{withx}>\mathrm{0} \\ $$

Question Number 44471 Answers: 0 Comments: 2

calculste ∫_0 ^∞ ((ln(x))/((1+x)^2 ))dx

$${calculste}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 44470 Answers: 0 Comments: 0

find ∫_0 ^∞ (dt/(1+t^2 sin^2 t))

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}} \\ $$

Question Number 44468 Answers: 0 Comments: 0

let u_n = Σ_(k=1) ^n (e^(−k) /(√k)) find a equivalent of u_n when n→+∞ .

$${let}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{e}^{−{k}} }{\sqrt{{k}}} \\ $$$${find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \:{when}\:{n}\rightarrow+\infty\:. \\ $$

Question Number 44467 Answers: 1 Comments: 0

calculate lim_(x→0) ((x^n sinx −sin(x^n ))/x) with n integr natural.

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{{x}^{{n}} {sinx}\:−{sin}\left({x}^{{n}} \right)}{{x}}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$

Question Number 44466 Answers: 0 Comments: 4

let f(x) = ∫_0 ^∞ ((x sinx)/(a^2 +x^4 ))dx with a>0 1) find a explicit form of f(a) 2) find g(a) = ∫_0 ^∞ ((xsinx)/((a^2 +x^4 )^2 ))dx 3)find the value of ∫_0 ^∞ ((x sinx)/(x^4 +1))dx 4) find the value of ∫_0 ^∞ ((xsinx)/((x^4 +1)^2 ))dx .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}\:{sinx}}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{4}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsinx}}{\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}\:{sinx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsinx}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:. \\ $$$$ \\ $$

Question Number 44454 Answers: 0 Comments: 4

Question Number 44444 Answers: 1 Comments: 0

simplify (√((4x^2 y)^(2/3) +(8x^2 y^2 )^4 ))

$${simplify}\:\:\:\:\sqrt{\left(\mathrm{4}{x}^{\mathrm{2}} {y}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\left(\mathrm{8}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$

Question Number 44441 Answers: 1 Comments: 0

Question Number 44436 Answers: 1 Comments: 1

Question Number 44430 Answers: 0 Comments: 0

Question Number 44424 Answers: 1 Comments: 0

by considering a sermicircle from −r to r prove that area of circle is πr^2

$${by}\:{considering}\:\:{a}\:{sermicircle}\:{from}\:−{r}\:{to}\:\:{r}\:{prove}\:{that}\:{area}\:{of}\:{circle}\:{is}\:\pi{r}^{\mathrm{2}} \\ $$

Question Number 44423 Answers: 1 Comments: 0

evaluate ∫3^x dx

$${evaluate}\:\int\mathrm{3}^{{x}} {dx} \\ $$

Question Number 44422 Answers: 0 Comments: 3

use substitution x=cos^2 θ+3sin^2 θ show that∫_1 ^3 (dx/(√((x−1)(3−x))))=π

$${use}\:{substitution}\:{x}=\mathrm{cos}\:^{\mathrm{2}} \theta+\mathrm{3}{sin}^{\mathrm{2}} \theta \\ $$$${show}\:{that}\int_{\mathrm{1}} ^{\mathrm{3}} \frac{{dx}}{\sqrt{\left({x}−\mathrm{1}\right)\left(\mathrm{3}−{x}\right)}}=\pi \\ $$

Question Number 44411 Answers: 0 Comments: 0

Question Number 44397 Answers: 2 Comments: 4

If x is nearly equal to 1 then ((mx^m −nx^n )/(m−n))=

$${If}\:{x}\:{is}\:{nearly}\:{equal}\:{to}\:\mathrm{1}\:{then} \\ $$$$\frac{{mx}^{{m}} −{nx}^{{n}} }{{m}−{n}}= \\ $$

Question Number 44395 Answers: 1 Comments: 3

Question Number 44389 Answers: 0 Comments: 0

Given 2 events A and B such that P(A)=(1/3) , P(A∪B)=(3/4) then find range of P(B)?

$${Given}\:\mathrm{2}\:{events}\:{A}\:{and}\:{B}\:{such}\:{that} \\ $$$${P}\left({A}\right)=\frac{\mathrm{1}}{\mathrm{3}}\:,\:{P}\left({A}\cup{B}\right)=\frac{\mathrm{3}}{\mathrm{4}}\:{then} \\ $$$${find}\:{range}\:{of}\:{P}\left({B}\right)? \\ $$

Question Number 44387 Answers: 0 Comments: 0

Question Number 44384 Answers: 2 Comments: 6

Let a and b are real numbers such that a > b > 0 Find the minimum value of (√2)a^3 + (3/(ab − b^2 ))

$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$${a}\:>\:{b}\:>\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\sqrt{\mathrm{2}}{a}^{\mathrm{3}} \:+\:\frac{\mathrm{3}}{{ab}\:−\:{b}^{\mathrm{2}} } \\ $$

Question Number 44374 Answers: 1 Comments: 0

Question Number 44373 Answers: 1 Comments: 0

Question Number 44369 Answers: 1 Comments: 1

The domain of f(x)=(√(cos(sinx))) +(1−x)^(−1) + sin^(−1) ((x^2 +1)/(2x)) is.........=.

$${The}\:{domain}\:{of}\: \\ $$$${f}\left({x}\right)=\sqrt{{cos}\left({sinx}\right)}\:+\left(\mathrm{1}−{x}\overset{−\mathrm{1}} {\right)}+\:\mathrm{sin}^{−\mathrm{1}} \frac{{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{x}} \\ $$$${is}.........=. \\ $$

Question Number 44365 Answers: 2 Comments: 1

Question Number 44356 Answers: 0 Comments: 0

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