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Question Number 31463 Answers: 0 Comments: 0

let give the function f(x)=∫_0 ^π ln(1+xcosθ)dθ with ∣x∣<1 1) find a simple form of f(x) 2)calculate ∫_0 ^π ln(1−cosθ)dθ 3)calculate ∫_0 ^π ln(1+cosθ)dθ.

$${let}\:{give}\:{the}\:{function}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}+{xcos}\theta\right){d}\theta\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\mathrm{1}−{cos}\theta\right){d}\theta \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}+{cos}\theta\right){d}\theta. \\ $$

Question Number 31462 Answers: 0 Comments: 0

find ∫_0 ^∞ ((arctan(x+(1/x)))/(1+x^2 ))dx.

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

Question Number 31461 Answers: 0 Comments: 0

let give u_n = Σ_(k=1) ^n sin ((k/n)) sin((k/n^2 )) 1) prove that the sequence (u_n ) is convergent 2) find lim_(n→∞) u_n .

$${let}\:{give}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\:\left(\frac{{k}}{{n}}\right)\:{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{sequence}\:\left({u}_{{n}} \right)\:{is}\:{convergent} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} . \\ $$

Question Number 31460 Answers: 0 Comments: 1

find in terms of n the value of A_n = ∫_0 ^1 Π_(k=1) ^(n−1) (x^2 −2xcos(((kπ)/n)) +1)dx with n from N^★ .

$${find}\:{in}\:{terms}\:{of}\:\:{n}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \left({x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\left(\frac{{k}\pi}{{n}}\right)\:+\mathrm{1}\right){dx}\:\:\:{with}\:{n}\:{from}\:{N}^{\bigstar} . \\ $$

Question Number 31456 Answers: 0 Comments: 2

Find sum of S= (2/3) + (4/3^2 ) + (6/3^3 ) + (8/3^4 ) +......+∞ ?

$$\mathbb{F}{ind}\:{sum}\:{of} \\ $$$${S}=\:\frac{\mathrm{2}}{\mathrm{3}}\:+\:\frac{\mathrm{4}}{\mathrm{3}^{\mathrm{2}} }\:+\:\frac{\mathrm{6}}{\mathrm{3}^{\mathrm{3}} }\:+\:\frac{\mathrm{8}}{\mathrm{3}^{\mathrm{4}} }\:+......+\infty\:? \\ $$

Question Number 31459 Answers: 0 Comments: 0

find ∫_0 ^(π/4) ((cost)/(cos^3 t +sin^3 t)) dt.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{cost}}{{cos}^{\mathrm{3}} {t}\:+{sin}^{\mathrm{3}} {t}}\:{dt}. \\ $$

Question Number 31458 Answers: 0 Comments: 0

calculate ∫_0 ^(√3) arcsin(((2t)/(1+t^2 )))dt .

$${calculate}\:\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:\:{arcsin}\left(\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right){dt}\:. \\ $$

Question Number 31442 Answers: 1 Comments: 2

Question Number 31441 Answers: 1 Comments: 1

Question Number 31436 Answers: 0 Comments: 3

Question Number 31423 Answers: 1 Comments: 1

Question Number 31422 Answers: 0 Comments: 6

find Σ_(n=1) ^∞ arctan((2/(16n^2 +8n−2))).

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{arctan}\left(\frac{\mathrm{2}}{\mathrm{16}{n}^{\mathrm{2}} \:+\mathrm{8}{n}−\mathrm{2}}\right). \\ $$

Question Number 31421 Answers: 1 Comments: 0

find the value of Σ_(n=1) ^∞ arctan((3/(9n^2 −3n−1))).

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} {arctan}\left(\frac{\mathrm{3}}{\mathrm{9}{n}^{\mathrm{2}} −\mathrm{3}{n}−\mathrm{1}}\right). \\ $$

Question Number 31419 Answers: 0 Comments: 0

find ∫_0 ^∞ (((1+t^2 )arctant)/(1+t^4 ))dt .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}+{t}^{\mathrm{2}} \right){arctant}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\:. \\ $$

Question Number 31418 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctanx)/(x^2 +x+1)) dx.

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctanx}}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}\:{dx}. \\ $$

Question Number 31417 Answers: 0 Comments: 0

let give u_n = Σ_(k=1) ^n (1/k) −ln(n) 1) prove that u_n is convergent 2) if γ=lim_(n→∞) u_n prove the 0<γ<1 .

$${let}\:{give}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:−{ln}\left({n}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{u}_{{n}} \:{is}\:{convergent}\: \\ $$$$\left.\mathrm{2}\right)\:{if}\:\:\:\gamma={lim}_{{n}\rightarrow\infty} {u}_{{n}} \:\:{prove}\:{the}\:\mathrm{0}<\gamma<\mathrm{1}\:. \\ $$

Question Number 31416 Answers: 0 Comments: 0

find the value of Σ_(n=2) ^∞ (1/(n^4 −1)) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{4}} −\mathrm{1}}\:\:. \\ $$

Question Number 31415 Answers: 0 Comments: 0

let 0<x<1 find f(x)=∫_0 ^x lnt .ln(1−t)dt.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {lnt}\:.{ln}\left(\mathrm{1}−{t}\right){dt}. \\ $$

Question Number 31414 Answers: 0 Comments: 0

find ∫_0 ^((√3) ) (dx/(x^2 +(√(x+1)))) .

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

Question Number 31409 Answers: 1 Comments: 0

The maximum area of the triangle whose sides a,b and c satisfy 0≤a≤1 , 1≤b≤2 , 2≤c≤3 is : A) 1 B) 2 C) 1.5 D) 0.5 ?

$${The}\:{maximum}\:{area}\:{of}\:{the}\:{triangle} \\ $$$${whose}\:{sides}\:{a},{b}\:{and}\:{c}\:{satisfy}\: \\ $$$$\mathrm{0}\leqslant{a}\leqslant\mathrm{1}\:,\:\mathrm{1}\leqslant{b}\leqslant\mathrm{2}\:,\:\mathrm{2}\leqslant{c}\leqslant\mathrm{3}\:{is}\:: \\ $$$$\left.{A}\right)\:\mathrm{1} \\ $$$$\left.{B}\right)\:\mathrm{2} \\ $$$$\left.{C}\right)\:\mathrm{1}.\mathrm{5} \\ $$$$\left.{D}\right)\:\mathrm{0}.\mathrm{5}\:\:\:\:\:\:\:? \\ $$

Question Number 31406 Answers: 0 Comments: 2

Question Number 31383 Answers: 1 Comments: 6

Question Number 31382 Answers: 1 Comments: 0

Question Number 31371 Answers: 1 Comments: 0

A point moves in xy plane such that sum of its distance from two mutually perpendicular lines is always 3.The area encloded by the locus of the point is

$${A}\:{point}\:{moves}\:{in}\:{xy}\:{plane}\:{such}\: \\ $$$${that}\:{sum}\:{of}\:{its}\:{distance}\:{from}\:{two}\: \\ $$$${mutually}\:{perpendicular}\:{lines}\:{is} \\ $$$${always}\:\mathrm{3}.{The}\:{area}\:{encloded}\:{by} \\ $$$${the}\:{locus}\:{of}\:{the}\:{point}\:{is} \\ $$

Question Number 31369 Answers: 0 Comments: 0

Question Number 31350 Answers: 0 Comments: 0

Are centripetal and centrifugal forces always equal?If No,then in what conditions are they equal or not equal?

$${Are}\:{centripetal}\:{and}\:{centrifugal} \\ $$$${forces}\:{always}\:{equal}?{If}\:{No},{then} \\ $$$${in}\:{what}\:{conditions}\:{are}\:{they}\:{equal} \\ $$$${or}\:{not}\:{equal}? \\ $$

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