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Question Number 140488 Answers: 0 Comments: 2

Question Number 140487 Answers: 0 Comments: 0

∫_0 ^∞ ((1/( (√(1+x))))−(1/( (√(1+x^2 )))))(dx/x)=log(2)

$$\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\right)\frac{{dx}}{{x}}={log}\left(\mathrm{2}\right) \\ $$

Question Number 140481 Answers: 0 Comments: 0

Question Number 140475 Answers: 0 Comments: 0

∫_0 ^( (π/2)) ((tan^(−1) ((√(tanx))))/(tanx))dx how can solve this

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{{tan}^{−\mathrm{1}} \left(\sqrt{{tanx}}\right)}{{tanx}}{dx} \\ $$$${how}\:{can}\:{solve}\:{this} \\ $$$$ \\ $$

Question Number 140474 Answers: 1 Comments: 0

Question Number 140447 Answers: 1 Comments: 0

Question Number 140446 Answers: 0 Comments: 0

If the area of a triangle with vertices Z_1 , Z_2 and Z_3 is the absolute value of the number λi determinant ((Z_1 ,Z_1 ^ ,1),(Z_2 ,Z_2 ^ ,1),(Z_3 ,Z_3 ^ ,1)) then the value of 1/λ is equal to _____.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{with}\:\mathrm{vertices}\:{Z}_{\mathrm{1}} ,\:{Z}_{\mathrm{2}} \:\mathrm{and}\:{Z}_{\mathrm{3}} \:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{absolute}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\lambda{i}\:\:\begin{vmatrix}{{Z}_{\mathrm{1}} }&{\bar {{Z}}_{\mathrm{1}} }&{\mathrm{1}}\\{{Z}_{\mathrm{2}} }&{\bar {{Z}}_{\mathrm{2}} }&{\mathrm{1}}\\{{Z}_{\mathrm{3}} }&{\bar {{Z}}_{\mathrm{3}} }&{\mathrm{1}}\end{vmatrix} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{1}/\lambda\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\_\_\_\_\_. \\ $$

Question Number 140445 Answers: 0 Comments: 0

If the straight lines a_i ^ z+a_i z^ +b_i =0(i=1, 2, 3), where b_i are real, are concurrent, then Σb_i (a_2 a_3 ^ −a_2 ^ a_3 ) is equal to _____.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{straight}\:\mathrm{lines}\:\bar {{a}}_{{i}} {z}+{a}_{{i}} \bar {{z}}+{b}_{{i}} =\mathrm{0}\left({i}=\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right),\:\mathrm{where} \\ $$$${b}_{{i}} \:\mathrm{are}\:\mathrm{real},\:\mathrm{are}\:\mathrm{concurrent},\:\mathrm{then}\:\Sigma{b}_{{i}} \left({a}_{\mathrm{2}} \bar {{a}}_{\mathrm{3}} −\bar {{a}}_{\mathrm{2}} {a}_{\mathrm{3}} \right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\_\_\_\_\_. \\ $$

Question Number 140444 Answers: 1 Comments: 0

If the points 1+2i and −1+4i are reflections of each other in the line z(1+i)+z^ (1−i)+K=0, then the value of K is _____.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{points}\:\mathrm{1}+\mathrm{2}{i}\:\mathrm{and}\:−\mathrm{1}+\mathrm{4}{i}\:\mathrm{are}\:\mathrm{reflections}\:\mathrm{of} \\ $$$$\mathrm{each}\:\mathrm{other}\:\mathrm{in}\:\mathrm{the}\:\mathrm{line}\:{z}\left(\mathrm{1}+{i}\right)+\bar {{z}}\left(\mathrm{1}−{i}\right)+{K}=\mathrm{0},\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{K}\:\mathrm{is}\:\_\_\_\_\_. \\ $$

Question Number 140443 Answers: 1 Comments: 0

If z_2 /z_1 is purely imaginary and a and b are non-zero real numbers, then ∣(az_1 +bz_2 )/(az_1 −bz_2 )∣ is equal to _____.

$$\mathrm{If}\:{z}_{\mathrm{2}} /{z}_{\mathrm{1}} \:\mathrm{is}\:\mathrm{purely}\:\mathrm{imaginary}\:\mathrm{and}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{non}-\mathrm{zero}\:\mathrm{real} \\ $$$$\mathrm{numbers},\:\mathrm{then}\:\mid\left({az}_{\mathrm{1}} +{bz}_{\mathrm{2}} \right)/\left({az}_{\mathrm{1}} −{bz}_{\mathrm{2}} \right)\mid\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\_\_\_\_\_. \\ $$

Question Number 140442 Answers: 1 Comments: 0

If z_1 and z_2 are complex numbers such that ∣z_2 ∣≠1 and ∣(z_1 −2z_2 )/(2−z_1 z_2 ^ )∣=1, then ∣z_1 ∣ is equal to _____.

$$\mathrm{If}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{2}} \mid\neq\mathrm{1}\:\mathrm{and} \\ $$$$\mid\left({z}_{\mathrm{1}} −\mathrm{2}{z}_{\mathrm{2}} \right)/\left(\mathrm{2}−{z}_{\mathrm{1}} \bar {{z}}_{\mathrm{2}} \right)\mid=\mathrm{1},\:\mathrm{then}\:\mid{z}_{\mathrm{1}} \mid\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\_\_\_\_\_. \\ $$

Question Number 140428 Answers: 1 Comments: 1

If 4x=3(Mod 6), find the first four values of x.

$$\mathrm{If}\:\mathrm{4}{x}=\mathrm{3}\left(\mathrm{Mod}\:\mathrm{6}\right),\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{four}\:\mathrm{values}\:\mathrm{of}\:{x}. \\ $$

Question Number 140477 Answers: 3 Comments: 0

If sin θ+sin^2 θ+sin^3 θ+.... = cos θ and 0<θ<(π/2) then find θ.

$${If}\:\:\mathrm{sin}\:\theta+\mathrm{sin}\:^{\mathrm{2}} \theta+\mathrm{sin}\:^{\mathrm{3}} \theta+.... \\ $$$$\:\:\:\:\:\:=\:\mathrm{cos}\:\theta\:\:\:\:\:{and}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}}\:{then} \\ $$$${find}\:\theta. \\ $$

Question Number 140424 Answers: 0 Comments: 0

Question Number 140494 Answers: 1 Comments: 0

tan^(−1) (((1−x)/(1+x)))+cot^(−1) (((1+x)/(1−x)))= (π/2) x=?

$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}\right)+\mathrm{cot}^{−\mathrm{1}} \left(\frac{\mathrm{1}+\mathrm{x}}{\mathrm{1}−\mathrm{x}}\right)=\:\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{x}=? \\ $$

Question Number 140417 Answers: 1 Comments: 0

if minimum value of g(a;b)=(√(a^2 +b^2 −10a−10b+50))+(√(b^2 −4y+20))+ +(√(a^2 −14a+74)) is n and occurs at a=γ , b=δ, the find (n+4γ+3δ)=?

$${if}\:{minimum}\:{value}\:{of} \\ $$$${g}\left({a};{b}\right)=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −\mathrm{10}{a}−\mathrm{10}{b}+\mathrm{50}}+\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{y}+\mathrm{20}}+ \\ $$$$+\sqrt{{a}^{\mathrm{2}} −\mathrm{14}{a}+\mathrm{74}} \\ $$$${is}\:{n}\:{and}\:{occurs}\:{at}\:{a}=\gamma\:,\:{b}=\delta,\:{the}\:{find} \\ $$$$\left({n}+\mathrm{4}\gamma+\mathrm{3}\delta\right)=? \\ $$

Question Number 140407 Answers: 0 Comments: 0

prove that ⟨ X:=R , τ_e ⟩ is a second topology space . τ_e is Euclidian topology on R. Hint :: B= { (r−(1/n) ,r+(1/n))∣ r∈Q , n∈N} is a base for τ_(e ) .....

$$\:\:\:\: \\ $$$$\:\:\:\:{prove}\:{that}\:\langle\:\mathrm{X}:=\mathbb{R}\:,\:\tau_{{e}} \:\rangle\:{is} \\ $$$$\:\:\:\:\:\:{a}\:{second}\:{topology}\:{space}\:. \\ $$$$\:\:\:\:\:\tau_{{e}} \:\:{is}\:\mathscr{E}{uclidian}\:{topology}\:{on}\:\mathbb{R}. \\ $$$$\:\:\:\mathrm{H}{int}\:::\:\:\mathcal{B}=\:\left\{\:\left({r}−\frac{\mathrm{1}}{{n}}\:,{r}+\frac{\mathrm{1}}{{n}}\right)\mid\:{r}\in\mathrm{Q}\:,\:{n}\in\mathbb{N}\right\} \\ $$$$\:\:\:{is}\:\:{a}\:{base}\:{for}\:\tau_{{e}\:} ..... \\ $$

Question Number 140405 Answers: 1 Comments: 0

find the value of :: Θ :=Σ_(n=1 ) ^∞ (1/(4n.(4n+1).(4n+2).(4n+3)))=?

$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:{find}\:\:{the}\:\:{value}\:{of}\::: \\ $$$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\Theta\::=\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{4}{n}.\left(\mathrm{4}{n}+\mathrm{1}\right).\left(\mathrm{4}{n}+\mathrm{2}\right).\left(\mathrm{4}{n}+\mathrm{3}\right)}=? \\ $$$$\:\:\:\:\: \\ $$

Question Number 140401 Answers: 2 Comments: 0

evaluate :: Φ:=∫_0 ^( ∞) xe^(−(x^2 /4)) ln(x)dx = m.( π γ) find ” m ” ......

$$\:\:\: \\ $$$$\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\Phi:=\int_{\mathrm{0}} ^{\:\infty} {xe}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}} {ln}\left({x}\right){dx}\:=\:{m}.\left(\:\pi\:\gamma\right) \\ $$$$\:\:\:\:\:\:{find}\:\:\:''\:\:{m}\:\:''\:...... \\ $$$$ \\ $$

Question Number 140399 Answers: 1 Comments: 0

𝛏 :=∫_0 ^( ∞) ((e^(−x^2 ) −e^(−x) )/x) dx = k.γ find ” k ” ... γ := Euler constant....

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\xi}\::=\int_{\mathrm{0}} ^{\:\infty} \:\frac{{e}^{−{x}^{\mathrm{2}} } −{e}^{−{x}} }{{x}}\:{dx}\:=\:{k}.\gamma\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{find}\:\:''\:{k}\:\:''\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\gamma\::=\:\mathscr{E}{uler}\:{constant}.... \\ $$

Question Number 140395 Answers: 0 Comments: 0

Q135933

$${Q}\mathrm{135933} \\ $$

Question Number 140392 Answers: 0 Comments: 0

let f:[0;1]→R, prove that ∃x_0 ,x_1 ,x_2 ∈(0;1) such that ((f(x_0 ))/x_0 ^2 )+((f(x_1 ))/(2x_1 ^2 ))=3f(x_2 )

$${let}\:{f}:\left[\mathrm{0};\mathrm{1}\right]\rightarrow\mathbb{R},\:{prove}\:{that}\:\exists{x}_{\mathrm{0}} ,{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} \in\left(\mathrm{0};\mathrm{1}\right) \\ $$$${such}\:{that}\:\:\frac{{f}\left({x}_{\mathrm{0}} \right)}{{x}_{\mathrm{0}} ^{\mathrm{2}} }+\frac{{f}\left({x}_{\mathrm{1}} \right)}{\mathrm{2}{x}_{\mathrm{1}} ^{\mathrm{2}} }=\mathrm{3}{f}\left({x}_{\mathrm{2}} \right) \\ $$

Question Number 140388 Answers: 3 Comments: 0

∫_0 ^∞ ((ln x)/((x^2 +a^2 )^5 )) dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{ln}\:\mathrm{x}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} \right)^{\mathrm{5}} }\:\mathrm{dx}\: \\ $$

Question Number 140387 Answers: 1 Comments: 0

Question Number 140430 Answers: 0 Comments: 0

(1+2x)(x^2 +1)(x^4 +x^3 −2x^2 +5x+1)^2 ={(x^3 −2x^2 +x+1)(x+1)(x^2 +1) −x^2 (1+2x)(x+3)}^2 Any good non-zero real solution to this equation in the exact form with the help of a calculator, perhaps...(please help)

$$\left(\mathrm{1}+\mathrm{2}{x}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{4}} +{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$=\left\{\left({x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{1}\right)\right. \\ $$$$\left.\:\:\:\:\:\:\:−{x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2}{x}\right)\left({x}+\mathrm{3}\right)\right\}^{\mathrm{2}} \\ $$$${Any}\:{good}\:{non}-{zero}\:{real}\:{solution} \\ $$$${to}\:{this}\:{equation}\:{in}\:{the}\:{exact}\:{form} \\ $$$${with}\:{the}\:{help}\:{of}\:{a}\:{calculator}, \\ $$$${perhaps}...\left({please}\:{help}\right) \\ $$

Question Number 140384 Answers: 1 Comments: 0

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