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Question Number 46073 Answers: 3 Comments: 1

Question Number 46032 Answers: 1 Comments: 0

Find the sum: Σ_(k = 1) ^n tan^(−1) (((2k)/(2 + k^2 + k^4 ))) Answer: tan^(−1) (n^2 + n + 1) − (π/4)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}:\:\:\:\:\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{tan}^{−\mathrm{1}} \:\left(\frac{\mathrm{2k}}{\mathrm{2}\:+\:\mathrm{k}^{\mathrm{2}} \:+\:\mathrm{k}^{\mathrm{4}} }\right) \\ $$$$ \\ $$$$\mathrm{Answer}:\:\:\:\:\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{n}\:+\:\mathrm{1}\right)\:−\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 46029 Answers: 0 Comments: 1

Question Number 46025 Answers: 1 Comments: 0

Question Number 46020 Answers: 1 Comments: 0

Question Number 46014 Answers: 0 Comments: 0

Question Number 46012 Answers: 2 Comments: 0

the normal at any point of hyperbola meets the axes at E,F.find the locus of the midpoint of EF.

$$\mathrm{the}\:\mathrm{normal}\:\mathrm{at}\:\mathrm{any}\:\mathrm{point} \\ $$$$\mathrm{of}\:\mathrm{hyperbola}\:\mathrm{meets}\:\mathrm{the}\:\mathrm{axes} \\ $$$$\mathrm{at}\:\mathrm{E},\mathrm{F}.\mathrm{find}\:\mathrm{the}\:\mathrm{locus}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:\mathrm{EF}. \\ $$

Question Number 46011 Answers: 0 Comments: 0

P is any point on rectangular xy=c^2 show that the line joining P to the centre and the tangent at P are equally inclined to the assymptotes

Question Number 46024 Answers: 0 Comments: 1

Question Number 46007 Answers: 1 Comments: 0

Question Number 46003 Answers: 1 Comments: 1

Question Number 45994 Answers: 2 Comments: 0

Question Number 45993 Answers: 1 Comments: 1

Question Number 45992 Answers: 1 Comments: 0

Question Number 45982 Answers: 1 Comments: 1

Find the value(s) of a such that a^x ≥ax with a, x∈R.

$${Find}\:{the}\:{value}\left({s}\right)\:{of}\:{a}\:{such}\:{that} \\ $$$${a}^{{x}} \geqslant{ax}\:{with}\:{a},\:{x}\in{R}. \\ $$

Question Number 45980 Answers: 1 Comments: 1

Question Number 45976 Answers: 0 Comments: 1

find u_n = ∫_0 ^∞ e^(−n[x]) cos(nx)dx and v_n =∫_0 ^∞ e^(n[x]) sin(nx)dx 2) find nature of Σ u_n v_n and Σ (u_n /v_n )

$${find}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{n}\left[{x}\right]} {cos}\left({nx}\right){dx}\:{and}\:{v}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{e}^{{n}\left[{x}\right]} {sin}\left({nx}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} {v}_{{n}} \:\:{and}\:\Sigma\:\frac{{u}_{{n}} }{{v}_{{n}} } \\ $$

Question Number 45975 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(1+t^2 ))/(1+t^2 ))dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 45974 Answers: 0 Comments: 0

calculate A_n =∫_0 ^n e^(−n[x]) sin (2x)dx 2)study the cnvergence of Σ A_n

$${calculate}\:\:{A}_{{n}} =\int_{\mathrm{0}} ^{{n}} \:\:\:{e}^{−{n}\left[{x}\right]} \:{sin}\:\left(\mathrm{2}{x}\right){dx} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{cnvergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$

Question Number 45973 Answers: 0 Comments: 0

find ∫ sh(x)ln(x+(√(1+x^2 )))dx

$${find}\:\int\:\:{sh}\left({x}\right){ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 45972 Answers: 0 Comments: 0

find ∫ ch(x)ln(x+(√(x^2 −1)))dx

$${find}\:\:\int\:\:{ch}\left({x}\right){ln}\left({x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}\right){dx} \\ $$

Question Number 45971 Answers: 0 Comments: 0

calculate f(x)=∫_0 ^1 ((arctan(xt))/(1+x^2 t^2 ))dt

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} }{dt} \\ $$

Question Number 45970 Answers: 1 Comments: 1

find ∫ ((arcsin(2x))/(√(1−4x^2 )))dx

$${find}\:\int\:\:\frac{{arcsin}\left(\mathrm{2}{x}\right)}{\sqrt{\mathrm{1}−\mathrm{4}{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 45968 Answers: 1 Comments: 2

1)find Σ_(n=1) ^∞ ((cos(nx))/n) and Σ_(n=1) ^∞ ((sin(nx))/n) 2) calculate Σ_(n=1) ^∞ (1/n)cos(((2nπ)/3)) and Σ_(n=1) ^∞ (1/n)sin(((2nπ)/3))

$$\left.\mathrm{1}\right){find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{cos}\left({nx}\right)}{{n}}\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{sin}\left({nx}\right)}{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}}{cos}\left(\frac{\mathrm{2}{n}\pi}{\mathrm{3}}\right)\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}}{sin}\left(\frac{\mathrm{2}{n}\pi}{\mathrm{3}}\right) \\ $$

Question Number 45969 Answers: 0 Comments: 0

1) find f(x)=∫_0 ^1 ln(1+ix)dx 2) calculate f^′ (x)

$$\left.\mathrm{1}\right)\:{find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{ix}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$

Question Number 45961 Answers: 0 Comments: 0

let f_n (x)=(−1)^n ln(1+(x^2 /(n(1+x^2 )))) and f(x)=Σ f_n (x) find lim_(x→+∞) f(x).

$${let}\:{f}_{{n}} \left({x}\right)=\left(−\mathrm{1}\right)^{{n}} \:{ln}\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{{n}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\right)\:{and}\:{f}\left({x}\right)=\Sigma\:{f}_{{n}} \left({x}\right) \\ $$$${find}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$

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