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Question Number 31459 Answers: 0 Comments: 0
$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{cost}}{{cos}^{\mathrm{3}} {t}\:+{sin}^{\mathrm{3}} {t}}\:{dt}. \\ $$
Question Number 31458 Answers: 0 Comments: 0
$${calculate}\:\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:\:{arcsin}\left(\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right){dt}\:. \\ $$
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
$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{arctan}\left(\frac{\mathrm{2}}{\mathrm{16}{n}^{\mathrm{2}} \:+\mathrm{8}{n}−\mathrm{2}}\right). \\ $$
Question Number 31421 Answers: 1 Comments: 0
$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} {arctan}\left(\frac{\mathrm{3}}{\mathrm{9}{n}^{\mathrm{2}} −\mathrm{3}{n}−\mathrm{1}}\right). \\ $$
Question Number 31419 Answers: 0 Comments: 0
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}+{t}^{\mathrm{2}} \right){arctant}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\:. \\ $$
Question Number 31418 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctanx}}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}\:{dx}. \\ $$
Question Number 31417 Answers: 0 Comments: 0
$${let}\:{give}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:−{ln}\left({n}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{u}_{{n}} \:{is}\:{convergent}\: \\ $$$$\left.\mathrm{2}\right)\:{if}\:\:\:\gamma={lim}_{{n}\rightarrow\infty} {u}_{{n}} \:\:{prove}\:{the}\:\mathrm{0}<\gamma<\mathrm{1}\:. \\ $$
Question Number 31416 Answers: 0 Comments: 0
$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{4}} −\mathrm{1}}\:\:. \\ $$
Question Number 31415 Answers: 0 Comments: 0
$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {lnt}\:.{ln}\left(\mathrm{1}−{t}\right){dt}. \\ $$
Question Number 31414 Answers: 0 Comments: 0
$${find}\:\:\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}\:} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:\:+\sqrt{{x}+\mathrm{1}}}\:. \\ $$
Question Number 31409 Answers: 1 Comments: 0
$${The}\:{maximum}\:{area}\:{of}\:{the}\:{triangle} \\ $$$${whose}\:{sides}\:{a},{b}\:{and}\:{c}\:{satisfy}\: \\ $$$$\mathrm{0}\leqslant{a}\leqslant\mathrm{1}\:,\:\mathrm{1}\leqslant{b}\leqslant\mathrm{2}\:,\:\mathrm{2}\leqslant{c}\leqslant\mathrm{3}\:{is}\:: \\ $$$$\left.{A}\right)\:\mathrm{1} \\ $$$$\left.{B}\right)\:\mathrm{2} \\ $$$$\left.{C}\right)\:\mathrm{1}.\mathrm{5} \\ $$$$\left.{D}\right)\:\mathrm{0}.\mathrm{5}\:\:\:\:\:\:\:? \\ $$
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
$${A}\:{point}\:{moves}\:{in}\:{xy}\:{plane}\:{such}\: \\ $$$${that}\:{sum}\:{of}\:{its}\:{distance}\:{from}\:{two}\: \\ $$$${mutually}\:{perpendicular}\:{lines}\:{is} \\ $$$${always}\:\mathrm{3}.{The}\:{area}\:{encloded}\:{by} \\ $$$${the}\:{locus}\:{of}\:{the}\:{point}\:{is} \\ $$
Question Number 31369 Answers: 0 Comments: 0
Question Number 31350 Answers: 0 Comments: 0
$${Are}\:{centripetal}\:{and}\:{centrifugal} \\ $$$${forces}\:{always}\:{equal}?{If}\:{No},{then} \\ $$$${in}\:{what}\:{conditions}\:{are}\:{they}\:{equal} \\ $$$${or}\:{not}\:{equal}? \\ $$
Question Number 31340 Answers: 0 Comments: 0
$${Deduce}\:{the}\:{power}\:{series}\:{of}\:{sin}^{\mathrm{2}} {x}. \\ $$$${Hence}\:{show}\:{that}\:{if}\:{x}\:{is}\:{small}\:{then} \\ $$$$\left({sin}^{\mathrm{2}} {x}\:−\:{x}^{\mathrm{2}} {cosx}\right)/{x}^{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{6}}\:−\:\frac{{x}^{\mathrm{2}} }{\mathrm{360}} \\ $$
Question Number 31336 Answers: 0 Comments: 1
$${Find}\:{the}\:{principal}\:{value}\:{of} \\ $$$${z}=\left(\mathrm{1}−{i}\right)^{\mathrm{1}+{i}} .{Hence}\:{find}\:{the} \\ $$$${modulus}\:{of}\:{the}\:{result}. \\ $$
Question Number 31335 Answers: 0 Comments: 3
$${Find}\:{the}\:{pricipal}\:{value}\:{of}\: \\ $$$${z}=\left(\mathrm{1}−{i}\right)^{{i}} \\ $$
Question Number 31329 Answers: 0 Comments: 5
Question Number 31327 Answers: 0 Comments: 1
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