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Question Number 45976 Answers: 0 Comments: 1
$${find}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{n}\left[{x}\right]} {cos}\left({nx}\right){dx}\:{and}\:{v}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{e}^{{n}\left[{x}\right]} {sin}\left({nx}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} {v}_{{n}} \:\:{and}\:\Sigma\:\frac{{u}_{{n}} }{{v}_{{n}} } \\ $$
Question Number 45975 Answers: 0 Comments: 0
$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$
Question Number 45974 Answers: 0 Comments: 0
$${calculate}\:\:{A}_{{n}} =\int_{\mathrm{0}} ^{{n}} \:\:\:{e}^{−{n}\left[{x}\right]} \:{sin}\:\left(\mathrm{2}{x}\right){dx} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{cnvergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$
Question Number 45973 Answers: 0 Comments: 0
$${find}\:\int\:\:{sh}\left({x}\right){ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx} \\ $$
Question Number 45972 Answers: 0 Comments: 0
$${find}\:\:\int\:\:{ch}\left({x}\right){ln}\left({x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}\right){dx} \\ $$
Question Number 45971 Answers: 0 Comments: 0
$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} }{dt} \\ $$
Question Number 45970 Answers: 1 Comments: 1
$${find}\:\int\:\:\frac{{arcsin}\left(\mathrm{2}{x}\right)}{\sqrt{\mathrm{1}−\mathrm{4}{x}^{\mathrm{2}} }}{dx} \\ $$
Question Number 45968 Answers: 1 Comments: 2
$$\left.\mathrm{1}\right){find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{cos}\left({nx}\right)}{{n}}\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{sin}\left({nx}\right)}{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}}{cos}\left(\frac{\mathrm{2}{n}\pi}{\mathrm{3}}\right)\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}}{sin}\left(\frac{\mathrm{2}{n}\pi}{\mathrm{3}}\right) \\ $$
Question Number 45969 Answers: 0 Comments: 0
$$\left.\mathrm{1}\right)\:{find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{ix}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$
Question Number 45961 Answers: 0 Comments: 0
$${let}\:{f}_{{n}} \left({x}\right)=\left(−\mathrm{1}\right)^{{n}} \:{ln}\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{{n}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\right)\:{and}\:{f}\left({x}\right)=\Sigma\:{f}_{{n}} \left({x}\right) \\ $$$${find}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$
Question Number 45960 Answers: 1 Comments: 0
$${find}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{x}^{{n}} {sin}\left({nx}\right)}{{n}} \\ $$
Question Number 45957 Answers: 1 Comments: 1
$${let}\:{u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}!\left({n}−{k}\right)!}\:\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} {u}_{{n}} \\ $$
Question Number 45962 Answers: 0 Comments: 0
$${study}\:{the}\:{convervence}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\sqrt{{n}+\mathrm{1}}−\sqrt{{n}}}{{nln}\left({n}+\mathrm{1}\right)} \\ $$
Question Number 45963 Answers: 1 Comments: 1
$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}}{\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$
Question Number 45941 Answers: 1 Comments: 0
$$\mathrm{Find}\:\mathrm{0}<\theta<\mathrm{2}\pi\:\mathrm{with}\:{x},{y}\:\in\mathbb{R} \\ $$$$ \\ $$$${x}\centerdot{sin}\theta={y}\centerdot{cos}\theta \\ $$
Question Number 45936 Answers: 0 Comments: 1
Question Number 45932 Answers: 1 Comments: 4
$$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\frac{\mathrm{1}.\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{2}.\mathrm{3}^{\mathrm{2}} \:+\:...\:+\:\mathrm{n}\left(\mathrm{n}\:+\:\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{1}^{\mathrm{2}} .\mathrm{2}\:+\:\mathrm{2}^{\mathrm{2}} .\mathrm{3}\:+\:...\:+\:\mathrm{n}^{\mathrm{2}} \left(\mathrm{n}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{3n}\:+\:\mathrm{5}}{\mathrm{3n}\:+\:\mathrm{1}} \\ $$
Question Number 45931 Answers: 0 Comments: 0
Question Number 45930 Answers: 1 Comments: 0
Question Number 45920 Answers: 1 Comments: 0
Question Number 45916 Answers: 1 Comments: 0
$$\int{f}\left({x}\right){dx}={f}\left(×\right)+{c} \\ $$
Question Number 45901 Answers: 0 Comments: 1
Question Number 45898 Answers: 1 Comments: 1
Question Number 45907 Answers: 1 Comments: 3
$$\mathrm{7}×\left(\mathrm{6}+\mathrm{x}\right)−\mathrm{10}=\mathrm{60}\:\:\:\:\:\mathrm{plz}\:\mathrm{help}\:\mathrm{me} \\ $$
Question Number 45905 Answers: 2 Comments: 0
Question Number 45896 Answers: 2 Comments: 2
$$\boldsymbol{{solve}}\:\boldsymbol{{for}}\:\boldsymbol{{x}} \\ $$$$\boldsymbol{{x}}^{\mathrm{4}} −\mathrm{4}\boldsymbol{{x}}+\mathrm{1}=\mathrm{0} \\ $$
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