Question and Answers Forum

All Questions   Topic List

AllQuestion and Answers: Page 1685

Question Number 34665    Answers: 0   Comments: 0

simplify sin (2arcsinx)

$${simplify}\:\:{sin}\:\left(\mathrm{2}{arcsinx}\right) \\ $$

Question Number 34664    Answers: 0   Comments: 0

simplify g(x)= arctan((1/(2x^2 ))) −arctan((x/(x+1))) +arctan(((x−1)/x))

$${simplify} \\ $$$${g}\left({x}\right)=\:{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)\:−{arctan}\left(\frac{{x}}{{x}+\mathrm{1}}\right)\:+{arctan}\left(\frac{{x}−\mathrm{1}}{{x}}\right) \\ $$

Question Number 34663    Answers: 0   Comments: 0

simplify f(x)=arcsin((√(1−x^2 ))) −arctan((√((1−x)/(1+x))))

$${simplify}\: \\ $$$${f}\left({x}\right)={arcsin}\left(\sqrt{\left.\mathrm{1}−{x}^{\mathrm{2}} \right)}\:\:−{arctan}\left(\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}\right)\right. \\ $$

Question Number 34662    Answers: 0   Comments: 0

calculate I(a) =∫_(1/a) ^a ((ln(x))/(1+x^2 )) dx with a>0 2) calculate ∫_0 ^(+∞) ((ln(x))/(1+x^2 )) dx .

$${calculate}\:{I}\left({a}\right)\:\:=\int_{\frac{\mathrm{1}}{{a}}} ^{{a}} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 34661    Answers: 0   Comments: 0

let f(x)= ∫_0 ^1 (e^(−(1+t^2 )x) /(1+t^2 )) dt find a simple form of f(x)

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{−\left(\mathrm{1}+{t}^{\mathrm{2}} \right){x}} }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{find}\:{a}\:{simple}\:{form}\:{of} \\ $$$${f}\left({x}\right) \\ $$

Question Number 34653    Answers: 0   Comments: 0

Question Number 34647    Answers: 2   Comments: 0

if xsin^3 θ + ycos^3 θ=sinθcosθ and xsinθ −ycosθ=0 prove that x^2 + y^2 =1

$${if}\:\:\:{xsin}^{\mathrm{3}} \theta\:+\:{ycos}^{\mathrm{3}} \theta={sin}\theta{cos}\theta \\ $$$${and}\:{xsin}\theta\:−{ycos}\theta=\mathrm{0} \\ $$$${prove}\:{that}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} =\mathrm{1} \\ $$

Question Number 34646    Answers: 1   Comments: 0

show that 2tan^(−1) 2 + tan^(−1) 3= Π +tan^(−1) (1/3)

$${show}\:{that}\:\mathrm{2}{tan}^{−\mathrm{1}} \mathrm{2}\:+\:{tan}^{−\mathrm{1}} \mathrm{3}= \\ $$$$\Pi\:+{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 34645    Answers: 0   Comments: 0

show that 2tan^(−1) 2 + tan^(−1) 3= Π +tan^(−1) (1/3)

$${show}\:{that}\:\mathrm{2}{tan}^{−\mathrm{1}} \mathrm{2}\:+\:{tan}^{−\mathrm{1}} \mathrm{3}= \\ $$$$\Pi\:+{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 34639    Answers: 2   Comments: 0

Question Number 34635    Answers: 2   Comments: 4

calculate A(α) = ∫_0 ^1 ln(1+αix)dx 2) calculate ∫_0 ^1 ln(1+ix) dx (i^2 =−1)

$${calculate}\:{A}\left(\alpha\right)\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+\alpha{ix}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{ix}\right)\:{dx}\:\:\:\:\left({i}^{\mathrm{2}} \:=−\mathrm{1}\right) \\ $$

Question Number 34634    Answers: 1   Comments: 1

let f(x)=ln(1+ix) with ∣x∣<1 1) extract Re(f(x)) and Im(f(x)) 2) developp f(x) at integr serie.

$${let}\:{f}\left({x}\right)={ln}\left(\mathrm{1}+{ix}\right)\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({f}\left({x}\right)\right)\:{and}\:{Im}\left({f}\left({x}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\left({x}\right)\:{at}\:{integr}\:{serie}. \\ $$

Question Number 34633    Answers: 0   Comments: 0

let f(α) = ∫_(−∞) ^(+∞) ((arctan(1+αxi))/(1+x^2 ))dx find f(α) .

$${let}\:{f}\left(\alpha\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{1}+\alpha{xi}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$${find}\:{f}\left(\alpha\right)\:. \\ $$

Question Number 34632    Answers: 0   Comments: 0

let z∈C developp at integrserie f(z)=ln(1+z) with ∣z∣<1 . 2) give ln(2+i) at form of serie.

$${let}\:{z}\in{C}\:\:{developp}\:{at}\:{integrserie} \\ $$$${f}\left({z}\right)={ln}\left(\mathrm{1}+{z}\right)\:\:{with}\:\mid{z}\mid<\mathrm{1}\:. \\ $$$$\left.\mathrm{2}\right)\:{give}\:{ln}\left(\mathrm{2}+{i}\right)\:{at}\:{form}\:{of}\:{serie}. \\ $$

Question Number 34617    Answers: 2   Comments: 2

Question Number 34615    Answers: 0   Comments: 0

decompose inside R(x) the fraction F(x)= (x^2 /((x+1)^5 ( x+3)^8 ))

$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}+\mathrm{1}\right)^{\mathrm{5}} \left(\:{x}+\mathrm{3}\right)^{\mathrm{8}} } \\ $$

Question Number 34608    Answers: 0   Comments: 0

Question Number 34607    Answers: 0   Comments: 0

let p∈C[x] degp=n (x_i )_(1≤k≤n) the roots of p(x) a∈C?/p(a)≠0 1) calculate S_1 = Σ_(k=1) ^n (1/(x_k −a)) interms of p,p^′ and a 2)calculste S_2 =Σ_(k=1) ^n (1/((x_k −a)^2 )) interms of p,p^, p^(′′) and a.

$${let}\:{p}\in{C}\left[{x}\right]\:{degp}={n}\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} {the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$${a}\in{C}?/{p}\left({a}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{S}_{\mathrm{1}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}_{{k}} −{a}}\:{interms}\:{of}\:{p},{p}^{'} \:{and}\:{a} \\ $$$$\left.\mathrm{2}\right){calculste}\:{S}_{\mathrm{2}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({x}_{{k}} −{a}\right)^{\mathrm{2}} }\:\:{interms}\:{of}\:{p},{p}^{,} \\ $$$${p}^{''} \:{and}\:{a}. \\ $$

Question Number 34606    Answers: 0   Comments: 0

let give p(x)=(x+1)^n −(x−1)^n 1) factorize p(x) inside C[x] 2) find the value of Π_(k=1) ^p cotan(((kπ)/(2p+1)))

$${let}\:{give}\:{p}\left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} \:−\left({x}−\mathrm{1}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:\:{cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right) \\ $$

Question Number 34605    Answers: 0   Comments: 0

decompose inside R(x) thefraction F(x)= ((x^5 +1)/(x^2^ (x−1)^2 )) .

$${decompose}\:{inside}\:{R}\left({x}\right)\:{thefraction} \\ $$$${F}\left({x}\right)=\:\:\frac{{x}^{\mathrm{5}} \:+\mathrm{1}}{{x}^{\mathrm{2}^{} } \left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 34604    Answers: 0   Comments: 0

let p(x)= x^n +x+1 ∈C[x] and z∈C/p(z)=0 prove that ∣z∣<2 .

$${let}\:\:{p}\left({x}\right)=\:{x}^{{n}} \:+{x}+\mathrm{1}\:\in{C}\left[{x}\right]\:{and}\:{z}\in{C}/{p}\left({z}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:\mid{z}\mid<\mathrm{2}\:. \\ $$

Question Number 34603    Answers: 0   Comments: 0

prove that ∀ p∈K[x] p(x) −x divide p(p(x))−x

$${prove}\:{that}\:\forall\:{p}\in{K}\left[{x}\right]\:{p}\left({x}\right)\:−{x}\:{divide}\:{p}\left({p}\left({x}\right)\right)−{x} \\ $$

Question Number 34602    Answers: 0   Comments: 0

simplify Σ_(k=0) ^n ((k/n) −α)^2 C_n ^k x^k (1−x)^(n−k ) α∈C.

$${simplify}\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\frac{{k}}{{n}}\:−\alpha\right)^{\mathrm{2}} {C}_{{n}} ^{{k}} \:{x}^{{k}} \left(\mathrm{1}−{x}\right)^{{n}−{k}\:} \\ $$$$\alpha\in{C}. \\ $$

Question Number 34596    Answers: 0   Comments: 0

1) prove that Σ_(k=1) ^n H_k =(n+1)H_n −n 2) prove that Σ_(k=1) ^n H_k ^2 =(n+1)H_n ^2 −(3n+1)H_n +2n H_n =Σ_(k=1) ^n (1/k) .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} =\left({n}+\mathrm{1}\right){H}_{{n}} \:−{n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:\:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:\:−\left(\mathrm{3}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}}\:. \\ $$

Question Number 34595    Answers: 0   Comments: 0

simplify Σ_(k=1) ^n (((−1)^(k−1) )/k) C_n ^k

$${simplify}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{C}_{{n}} ^{{k}} \\ $$

Question Number 34594    Answers: 0   Comments: 0

prove that Σ_(k=0) ^p (−1)^k C_n ^k =(−1)^p C_(n−1) ^p

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{p}} \left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$

  Pg 1680      Pg 1681      Pg 1682      Pg 1683      Pg 1684      Pg 1685      Pg 1686      Pg 1687      Pg 1688      Pg 1689   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com