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Question Number 34665 Answers: 0 Comments: 0
$${simplify}\:\:{sin}\:\left(\mathrm{2}{arcsinx}\right) \\ $$
Question Number 34664 Answers: 0 Comments: 0
$${simplify} \\ $$$${g}\left({x}\right)=\:{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)\:−{arctan}\left(\frac{{x}}{{x}+\mathrm{1}}\right)\:+{arctan}\left(\frac{{x}−\mathrm{1}}{{x}}\right) \\ $$
Question Number 34663 Answers: 0 Comments: 0
$${simplify}\: \\ $$$${f}\left({x}\right)={arcsin}\left(\sqrt{\left.\mathrm{1}−{x}^{\mathrm{2}} \right)}\:\:−{arctan}\left(\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}\right)\right. \\ $$
Question Number 34662 Answers: 0 Comments: 0
$${calculate}\:{I}\left({a}\right)\:\:=\int_{\frac{\mathrm{1}}{{a}}} ^{{a}} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$
Question Number 34661 Answers: 0 Comments: 0
$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{−\left(\mathrm{1}+{t}^{\mathrm{2}} \right){x}} }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{find}\:{a}\:{simple}\:{form}\:{of} \\ $$$${f}\left({x}\right) \\ $$
Question Number 34653 Answers: 0 Comments: 0
Question Number 34647 Answers: 2 Comments: 0
$${if}\:\:\:{xsin}^{\mathrm{3}} \theta\:+\:{ycos}^{\mathrm{3}} \theta={sin}\theta{cos}\theta \\ $$$${and}\:{xsin}\theta\:−{ycos}\theta=\mathrm{0} \\ $$$${prove}\:{that}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} =\mathrm{1} \\ $$
Question Number 34646 Answers: 1 Comments: 0
$${show}\:{that}\:\mathrm{2}{tan}^{−\mathrm{1}} \mathrm{2}\:+\:{tan}^{−\mathrm{1}} \mathrm{3}= \\ $$$$\Pi\:+{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{3}} \\ $$
Question Number 34645 Answers: 0 Comments: 0
Question Number 34639 Answers: 2 Comments: 0
Question Number 34635 Answers: 2 Comments: 4
$${calculate}\:{A}\left(\alpha\right)\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+\alpha{ix}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{ix}\right)\:{dx}\:\:\:\:\left({i}^{\mathrm{2}} \:=−\mathrm{1}\right) \\ $$
Question Number 34634 Answers: 1 Comments: 1
$${let}\:{f}\left({x}\right)={ln}\left(\mathrm{1}+{ix}\right)\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({f}\left({x}\right)\right)\:{and}\:{Im}\left({f}\left({x}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\left({x}\right)\:{at}\:{integr}\:{serie}. \\ $$
Question Number 34633 Answers: 0 Comments: 0
$${let}\:{f}\left(\alpha\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{1}+\alpha{xi}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$${find}\:{f}\left(\alpha\right)\:. \\ $$
Question Number 34632 Answers: 0 Comments: 0
$${let}\:{z}\in{C}\:\:{developp}\:{at}\:{integrserie} \\ $$$${f}\left({z}\right)={ln}\left(\mathrm{1}+{z}\right)\:\:{with}\:\mid{z}\mid<\mathrm{1}\:. \\ $$$$\left.\mathrm{2}\right)\:{give}\:{ln}\left(\mathrm{2}+{i}\right)\:{at}\:{form}\:{of}\:{serie}. \\ $$
Question Number 34617 Answers: 2 Comments: 2
Question Number 34615 Answers: 0 Comments: 0
$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}+\mathrm{1}\right)^{\mathrm{5}} \left(\:{x}+\mathrm{3}\right)^{\mathrm{8}} } \\ $$
Question Number 34608 Answers: 0 Comments: 0
Question Number 34607 Answers: 0 Comments: 0
$${let}\:{p}\in{C}\left[{x}\right]\:{degp}={n}\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} {the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$${a}\in{C}?/{p}\left({a}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{S}_{\mathrm{1}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}_{{k}} −{a}}\:{interms}\:{of}\:{p},{p}^{'} \:{and}\:{a} \\ $$$$\left.\mathrm{2}\right){calculste}\:{S}_{\mathrm{2}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({x}_{{k}} −{a}\right)^{\mathrm{2}} }\:\:{interms}\:{of}\:{p},{p}^{,} \\ $$$${p}^{''} \:{and}\:{a}. \\ $$
Question Number 34606 Answers: 0 Comments: 0
$${let}\:{give}\:{p}\left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} \:−\left({x}−\mathrm{1}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:\:{cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right) \\ $$
Question Number 34605 Answers: 0 Comments: 0
$${decompose}\:{inside}\:{R}\left({x}\right)\:{thefraction} \\ $$$${F}\left({x}\right)=\:\:\frac{{x}^{\mathrm{5}} \:+\mathrm{1}}{{x}^{\mathrm{2}^{} } \left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$
Question Number 34604 Answers: 0 Comments: 0
$${let}\:\:{p}\left({x}\right)=\:{x}^{{n}} \:+{x}+\mathrm{1}\:\in{C}\left[{x}\right]\:{and}\:{z}\in{C}/{p}\left({z}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:\mid{z}\mid<\mathrm{2}\:. \\ $$
Question Number 34603 Answers: 0 Comments: 0
$${prove}\:{that}\:\forall\:{p}\in{K}\left[{x}\right]\:{p}\left({x}\right)\:−{x}\:{divide}\:{p}\left({p}\left({x}\right)\right)−{x} \\ $$
Question Number 34602 Answers: 0 Comments: 0
$${simplify}\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\frac{{k}}{{n}}\:−\alpha\right)^{\mathrm{2}} {C}_{{n}} ^{{k}} \:{x}^{{k}} \left(\mathrm{1}−{x}\right)^{{n}−{k}\:} \\ $$$$\alpha\in{C}. \\ $$
Question Number 34596 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} =\left({n}+\mathrm{1}\right){H}_{{n}} \:−{n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:\:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:\:−\left(\mathrm{3}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}}\:. \\ $$
Question Number 34595 Answers: 0 Comments: 0
$${simplify}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{C}_{{n}} ^{{k}} \\ $$
Question Number 34594 Answers: 0 Comments: 0
$${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{p}} \left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$
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