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Question Number 101040 Answers: 1 Comments: 0

Express 2 sin θ cos 6θ in the form sin A − sin B (i) using that result prove that 2sin θ( cos 6θ + cos 4θ + cos 2θ) = sin 7θ−sin θ (ii) deduce the result cos (((12π)/7)) + cos (((8π)/7)) + cos (((4π)/7)) = −(1/2) (iii) hence find a general solution to ((sin7θ − sin θ)/(cos 6θ + cos 4θ + cos 2θ)) = 1

$$\:\mathrm{Express}\:\mathrm{2}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\mathrm{6}\theta\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\:\mathrm{sin}\:{A}\:−\:\mathrm{sin}\:{B} \\ $$$$\left({i}\right)\:\mathrm{using}\:\mathrm{that}\:\mathrm{result}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{2sin}\:\theta\left(\:\mathrm{cos}\:\mathrm{6}\theta\:+\:\mathrm{cos}\:\mathrm{4}\theta\:+\:\mathrm{cos}\:\mathrm{2}\theta\right)\:=\:\mathrm{sin}\:\mathrm{7}\theta−\mathrm{sin}\:\theta \\ $$$$\left({ii}\right)\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{result}\:\mathrm{cos}\:\left(\frac{\mathrm{12}\pi}{\mathrm{7}}\right)\:+\:\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{7}}\right)\:+\:\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{7}}\right)\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left({iii}\right)\:\mathrm{hence}\:\mathrm{find}\:\mathrm{a}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{to}\:\frac{\mathrm{sin7}\theta\:−\:\mathrm{sin}\:\theta}{\mathrm{cos}\:\mathrm{6}\theta\:+\:\mathrm{cos}\:\mathrm{4}\theta\:+\:\mathrm{cos}\:\mathrm{2}\theta}\:=\:\mathrm{1} \\ $$

Question Number 101079 Answers: 2 Comments: 2

Question Number 101082 Answers: 0 Comments: 2

The value of e^(log_(10) tan 1°+log_(10) tan 2°+log_(10) tan 3°+...+log_(10) tan 89°) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of} \\ $$$${e}^{\mathrm{log}_{\mathrm{10}} \:\mathrm{tan}\:\mathrm{1}°+\mathrm{log}_{\mathrm{10}} \:\mathrm{tan}\:\mathrm{2}°+\mathrm{log}_{\mathrm{10}} \:\mathrm{tan}\:\mathrm{3}°+...+\mathrm{log}_{\mathrm{10}} \:\mathrm{tan}\:\mathrm{89}°} \\ $$$$\mathrm{is} \\ $$

Question Number 101078 Answers: 1 Comments: 0

find the following sum Σ_(k=1) ^n C_( n) ^( k) x^(−k) (k−1)!

$${find}\:\:{the}\:\:{following}\:\:\:{sum} \\ $$$$\:\:\:\:\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{C}_{\:{n}} ^{\:{k}} {x}^{−{k}} \left({k}−\mathrm{1}\right)! \\ $$

Question Number 101026 Answers: 1 Comments: 0

lim_(x→∞) (x/e^( sinx −x) )

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}}{{e}^{\:\mathrm{sin}{x}\:−{x}} } \\ $$

Question Number 101023 Answers: 0 Comments: 0

∫tan^(1/5) x cotx secxdx

$$\int{tan}^{\frac{\mathrm{1}}{\mathrm{5}}} {x}\:{cotx}\:{secxdx} \\ $$

Question Number 101014 Answers: 0 Comments: 0

Show that ∫_(−∞) ^(+∞) (dx/(1+(x+tanx)^2 )) = π

$${Show}\:{that} \\ $$$$\int_{−\infty} ^{+\infty} \frac{{dx}}{\mathrm{1}+\left({x}+{tanx}\right)^{\mathrm{2}} }\:\:\:=\:\:\:\pi \\ $$

Question Number 101011 Answers: 0 Comments: 5

∫_0 ^∞ ((sinx)/x)dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}}{{x}}{dx} \\ $$

Question Number 101008 Answers: 1 Comments: 0

Given f(x) = { (((1/2)xe^(1/x) , x ≠ 0)),((0, x = 0 )) :} find (i) Thd domain of f (ii) check the continuity of f at x = 0 (iii) check its differentiability and its sign (i) sketch this curve and find lim_(x→−∞) f(x) and lim_(x→+∞) f(x)

$$\mathrm{Given}\:{f}\left({x}\right)\:=\:\begin{cases}{\frac{\mathrm{1}}{\mathrm{2}}{xe}^{\frac{\mathrm{1}}{{x}}} \:,\:{x}\:\neq\:\mathrm{0}}\\{\mathrm{0},\:{x}\:=\:\mathrm{0}\:}\end{cases} \\ $$$$\mathrm{find} \\ $$$$\left({i}\right)\:\mathrm{Thd}\:\mathrm{domain}\:\mathrm{of}\:{f} \\ $$$$\left({ii}\right)\:\mathrm{check}\:\mathrm{the}\:\mathrm{continuity}\:\mathrm{of}\:{f}\:\mathrm{at}\:{x}\:=\:\mathrm{0} \\ $$$$\left({iii}\right)\:\mathrm{check}\:\mathrm{its}\:\mathrm{differentiability}\:\mathrm{and}\:\mathrm{its}\:\mathrm{sign} \\ $$$$\left({i}\right)\:\mathrm{sketch}\:\mathrm{this}\:\mathrm{curve}\:\mathrm{and}\:\mathrm{find}\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:{f}\left({x}\right)\:\mathrm{and}\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}\right) \\ $$

Question Number 101018 Answers: 0 Comments: 0

∫_(−∞) ^∞ ((log(sin^2 x))/(1+x+e^x ))dx

$$\int_{−\infty} ^{\infty} \frac{{log}\left({sin}^{\mathrm{2}} {x}\right)}{\mathrm{1}+{x}+{e}^{{x}} }{dx} \\ $$

Question Number 101003 Answers: 0 Comments: 0

What is the value of x for wich the serie is converge? (1) Σ_(n≥0) x^((ln(n))/(n!)) ? (2) Σ_(n≥0 ) x^((ln(n!))/n) ?

$$\:\boldsymbol{{What}}\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{x}}\:\boldsymbol{{for}}\:\boldsymbol{{wich}}\:\boldsymbol{{the}}\:\boldsymbol{{serie}}\:\boldsymbol{{is}}\:\boldsymbol{{converge}}?\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\left(\mathrm{1}\right)\:\underset{\boldsymbol{{n}}\geqslant\mathrm{0}} {\sum}\boldsymbol{{x}}^{\frac{\boldsymbol{{ln}}\left(\boldsymbol{{n}}\right)}{\boldsymbol{{n}}!}} \:?\:\:\:\:\:\:\left(\mathrm{2}\right)\:\underset{\boldsymbol{{n}}\geqslant\mathrm{0}\:} {\sum}\boldsymbol{{x}}^{\frac{\boldsymbol{{ln}}\left(\boldsymbol{{n}}!\right)}{\boldsymbol{{n}}}} \:? \\ $$$$ \\ $$

Question Number 101001 Answers: 1 Comments: 0

Question Number 100988 Answers: 1 Comments: 1

4sin^2 x + sin 2x = 3 find solution set on x∈(0,2π)

$$\mathrm{4sin}\:^{\mathrm{2}} {x}\:+\:\mathrm{sin}\:\mathrm{2}{x}\:=\:\mathrm{3}\: \\ $$$${find}\:{solution}\:{set}\:{on}\:{x}\in\left(\mathrm{0},\mathrm{2}\pi\right) \\ $$

Question Number 100986 Answers: 0 Comments: 1

Question Number 100985 Answers: 0 Comments: 3

Question Number 100980 Answers: 0 Comments: 0

prove that ∫_(−∞) ^(+∞) ((1/(1+(x+tan(x))^2 ))dx)=𝛑

$$\:\:\:\:{prove}\:{that}\:\:\int_{−\infty} ^{+\infty} \left(\frac{\mathrm{1}}{\mathrm{1}+\left(\boldsymbol{{x}}+\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} }\boldsymbol{{dx}}\right)=\boldsymbol{\pi}\: \\ $$

Question Number 100976 Answers: 0 Comments: 1

find all possible values of x,y,z in terms of a,b,c gor a triplet (x,y,z) that satisfy x+(1/y)=a y+(1/z)=b z+(1/x)=c

$${find}\:{all}\:{possible}\:{values}\:{of}\:{x},{y},{z}\:{in}\:{terms} \\ $$$${of}\:{a},{b},{c}\:{gor}\:{a}\:{triplet}\:\left({x},{y},{z}\right)\:{that}\:{satisfy} \\ $$$$ \\ $$$${x}+\frac{\mathrm{1}}{{y}}={a} \\ $$$$ \\ $$$${y}+\frac{\mathrm{1}}{{z}}={b} \\ $$$$ \\ $$$${z}+\frac{\mathrm{1}}{{x}}={c} \\ $$

Question Number 100971 Answers: 1 Comments: 0

A woman sent 8 letters to her friends. The letters are kept in the addressed envelopes at random. The probability that 4 friends receive correct letters and 4 letters go to wrong destination, is ___

$$\mathrm{A}\:\mathrm{woman}\:\mathrm{sent}\:\mathrm{8}\:\mathrm{letters}\:\mathrm{to}\:\mathrm{her}\: \\ $$$$\mathrm{friends}.\:\mathrm{The}\:\mathrm{letters}\:\mathrm{are}\:\mathrm{kept}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{addressed}\:\mathrm{envelopes}\:\mathrm{at}\:\mathrm{random}.\: \\ $$$$\mathrm{The}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{4}\:\mathrm{friends}\: \\ $$$$\mathrm{receive}\:\mathrm{correct}\:\mathrm{letters}\:\mathrm{and}\:\mathrm{4}\:\mathrm{letters}\: \\ $$$$\mathrm{go}\:\mathrm{to}\:\mathrm{wrong}\:\mathrm{destination},\:\mathrm{is}\:\_\_\_\: \\ $$

Question Number 100969 Answers: 1 Comments: 0

find ∫_(−∞) ^∞ ((sin(cosx))/((x^2 −x+1)^2 ))dx

$$\mathrm{find}\:\int_{−\infty} ^{\infty} \:\:\frac{\mathrm{sin}\left(\mathrm{cosx}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 100968 Answers: 2 Comments: 0

Σ_(k=1) ^∞ (x+k)^(1/2^(k+1) ) =? x>0

$$\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\mathrm{x}+\mathrm{k}\right)^{\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{k}+\mathrm{1}} }} =?\:\:\:\:\:\:\mathrm{x}>\mathrm{0}\: \\ $$

Question Number 100967 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) ln(2+ sinθ)dθ

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{2}+\:\mathrm{sin}\theta\right)\mathrm{d}\theta \\ $$

Question Number 100966 Answers: 0 Comments: 3

find the fourier series of the function f(x)= { ((x −2≤x≤0)),((x+2 0≤x≤2)) :} help me sir ?