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AllQuestion and Answers: Page 1238

Question Number 90765 Answers: 0 Comments: 2

if ∫ (ln(x))^2 dx = x( ln^2 (x)+a ln(x)+b) +C a,b , C are constant. find the value of a and b

$${if}\:\int\:\left(\mathrm{ln}\left({x}\right)\right)^{\mathrm{2}} {dx}\:=\: \\ $$$${x}\left(\:\mathrm{ln}^{\mathrm{2}} \left({x}\right)+{a}\:\mathrm{ln}\left({x}\right)+{b}\right)\:+{C} \\ $$$${a},{b}\:,\:{C}\:{are}\:{constant}.\: \\ $$$${find}\:{the}\:{value}\:{of}\:{a}\:{and}\:{b}\: \\ $$

Question Number 90752 Answers: 1 Comments: 0

find ∫(1+x^2 )(√(1−x^2 ))dx

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

Question Number 90751 Answers: 0 Comments: 1

calculste lim_(n→∞) (1/n^2 )Σ_(k=1) ^n karctan((k/n))

$$\:{calculste}\:{lim}_{{n}\rightarrow\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\sum_{{k}=\mathrm{1}} ^{{n}} \:{karctan}\left(\frac{{k}}{{n}}\right) \\ $$

Question Number 90750 Answers: 0 Comments: 0

let α and β roots of x^2 −x+2=0 calculate A_n =Σ_(k=0) ^n (α^k +β^k ) B_n = Σ_(k=0) ^n (α^k −β^k )

$${let}\:\:\alpha\:{and}\:\beta\:{roots}\:{of}\:\:{x}^{\mathrm{2}} −{x}+\mathrm{2}=\mathrm{0}\:\:{calculate} \\ $$$${A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(\alpha^{{k}} \:+\beta^{{k}} \right) \\ $$$${B}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\alpha^{{k}} −\beta^{{k}} \right) \\ $$

Question Number 90749 Answers: 0 Comments: 1

find Σ_(k=0) ^n C_n ^k cos(((kπ)/n))

$${find}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}\left(\frac{{k}\pi}{{n}}\right) \\ $$

Question Number 90747 Answers: 1 Comments: 0

solve: t^(1/3) + t^(1/2) = 12

$$\mathrm{solve}:\:\:\:\mathrm{t}^{\mathrm{1}/\mathrm{3}} \:\:\:+\:\:\:\mathrm{t}^{\mathrm{1}/\mathrm{2}} \:\:\:=\:\:\:\mathrm{12} \\ $$

Question Number 90743 Answers: 1 Comments: 0

find nature of the serie Σ (−1)^n U_n with U_(n+1) =(e^(−U_n ) /(n+1)) (U_0 =1)

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:\left(−\mathrm{1}\right)^{{n}} \:{U}_{{n}} \\ $$$${with}\:\:{U}_{{n}+\mathrm{1}} =\frac{{e}^{−{U}_{{n}} } }{{n}+\mathrm{1}}\:\:\:\:\:\left({U}_{\mathrm{0}} =\mathrm{1}\right) \\ $$

Question Number 90739 Answers: 1 Comments: 2

solve for x and y tan^2 [π(x+y)]+cot^2 [π(x+y)] =1+(√((2x)/(x^2 +1)))

$${solve}\:{for}\:{x}\:{and}\:{y} \\ $$$$\mathrm{tan}\:^{\mathrm{2}} \left[\pi\left({x}+{y}\right)\right]+\mathrm{cot}\:^{\mathrm{2}} \left[\pi\left({x}+{y}\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{1}+\sqrt{\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$

Question Number 90730 Answers: 0 Comments: 0

let a ,b integer and C=a^2 +b^2 Prove that there exist a_n and b_n all integers such as C^n =a_n ^2 +b_n ^2 explicit a_5 and b_5 interm of a and b

$${let}\:{a}\:,{b}\:{integer}\:{and}\:\:{C}={a}^{\mathrm{2}} +{b}^{\mathrm{2}} \: \\ $$$${Prove}\:{that}\:{there}\:{exist}\:{a}_{{n}} \:{and}\:{b}_{{n}} \:{all}\: \\ $$$${integers}\:{such}\:{as}\:{C}^{{n}} ={a}_{{n}} ^{\mathrm{2}} \:+{b}_{{n}} ^{\mathrm{2}} \: \\ $$$${explicit}\:{a}_{\mathrm{5}} \:{and}\:{b}_{\mathrm{5}} \:{interm}\:{of}\:{a}\:{and}\:\:{b} \\ $$

Question Number 90727 Answers: 0 Comments: 0

Σ_(a,b≥1) (1/((a+b^2 )(a+b^2 +1))) nature and sum value

$$\underset{{a},{b}\geqslant\mathrm{1}} {\sum}\:\frac{\mathrm{1}}{\left({a}+{b}^{\mathrm{2}} \right)\left({a}+{b}^{\mathrm{2}} +\mathrm{1}\right)}\:\:\:\:\:\:{nature}\:{and}\:\:{sum}\:{value}\:\: \\ $$$$ \\ $$

Question Number 90722 Answers: 1 Comments: 3

lim_(x→0) ((((√(1+x)) −(√(1+x^2 )))^2 )/(cos x−1)) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\sqrt{\mathrm{1}+{x}}\:−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{\mathrm{2}} }{\mathrm{cos}\:{x}−\mathrm{1}}\:=\:? \\ $$

Question Number 90717 Answers: 0 Comments: 0

Prove that for all integer r≥2 HCF(r^n −1;r^m −1)=r^(HCF(n;m)) −1

$${Prove}\:{that}\:{for}\:{all}\:{integer}\:{r}\geqslant\mathrm{2}\: \\ $$$$\:{HCF}\left({r}^{{n}} −\mathrm{1};{r}^{{m}} −\mathrm{1}\right)={r}^{{HCF}\left({n};{m}\right)} −\mathrm{1} \\ $$

Question Number 90716 Answers: 0 Comments: 1

If x + (1/x) = 4 , what the value of ((x^6 −1)/x^3 )

$${If}\:{x}\:+\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{4}\:,\:{what}\:{the}\: \\ $$$${value}\:{of}\:\frac{{x}^{\mathrm{6}} −\mathrm{1}}{{x}^{\mathrm{3}} } \\ $$

Question Number 90709 Answers: 0 Comments: 2

α,β and γ are the roots of x^3 −9x+9=0 find the value of (1) α^(−3) +β^(−3) +γ^(−3) (2) α^(−5) +β^(−5) +γ^(−5)

$$\alpha,\beta\:{and}\:\gamma\:{are}\:{the}\:{roots}\:{of}\:\:{x}^{\mathrm{3}} −\mathrm{9}{x}+\mathrm{9}=\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:\left(\mathrm{1}\right)\:\alpha^{−\mathrm{3}} +\beta^{−\mathrm{3}} +\gamma^{−\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)\:\alpha^{−\mathrm{5}} +\beta^{−\mathrm{5}} +\gamma^{−\mathrm{5}} \\ $$

Question Number 90706 Answers: 2 Comments: 1

Question Number 90700 Answers: 0 Comments: 3

sin^2 (((7π)/8))+sin^2 (((3π)/8))+sin^2 (((5π)/8))+sin^2 ((π/8)) ?

$$\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\pi}{\mathrm{8}}\right)\:? \\ $$

Question Number 90698 Answers: 1 Comments: 3

how many solution the equation ⌊ x ⌋ +2016. {x} = 38?

$${how}\:{many}\:{solution}\:{the}\:{equation} \\ $$$$\lfloor\:{x}\:\rfloor\:+\mathrm{2016}.\:\left\{{x}\right\}\:=\:\mathrm{38}? \\ $$

Question Number 90692 Answers: 1 Comments: 2

Question Number 90679 Answers: 0 Comments: 1

if y=sin(msin^(−1) x), prove that (1−x^2 )y_(n+2) −(2n+1)xy_(n+1) +(m^2 −n^2 )y_n =0

$${if}\:{y}={sin}\left({m}\mathrm{sin}^{−\mathrm{1}} {x}\right),\:{prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}_{{n}+\mathrm{2}} −\left(\mathrm{2}{n}+\mathrm{1}\right){xy}_{{n}+\mathrm{1}} +\left({m}^{\mathrm{2}} −{n}^{\mathrm{2}} \right){y}_{{n}} =\mathrm{0} \\ $$

Question Number 92777 Answers: 1 Comments: 2

a_(n+1) =(2n+1)a_n a_1 =1 a_n =?

$$\mathrm{a}_{\mathrm{n}+\mathrm{1}} =\left(\mathrm{2n}+\mathrm{1}\right)\mathrm{a}_{\mathrm{n}} \\ $$$$\mathrm{a}_{\mathrm{1}} =\mathrm{1} \\ $$$$\mathrm{a}_{\mathrm{n}} =? \\ $$$$ \\ $$

Question Number 90661 Answers: 2 Comments: 0

show that (n^4 −n^2 ) is divisible by 12

$${show}\:{that}\:\left({n}^{\mathrm{4}} −{n}^{\mathrm{2}} \right)\:{is}\:{divisible}\:{by}\:\mathrm{12} \\ $$

Question Number 90647 Answers: 0 Comments: 14

Question Number 90641 Answers: 0 Comments: 4

Solve x^2 y′′+xy′+x^2 y=0

$${Solve}\:{x}^{\mathrm{2}} {y}''+{xy}'+{x}^{\mathrm{2}} {y}=\mathrm{0} \\ $$

Question Number 90637 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (1/2^n )tan((1/2^n ))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{{n}} }{tan}\left(\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\right) \\ $$

Question Number 90632 Answers: 1 Comments: 0

Question Number 90630 Answers: 1 Comments: 1

f(x) = xe^(−x) f^((2020)) (x) =

$${f}\left({x}\right)\:=\:{xe}^{−{x}} \\ $$$${f}^{\left(\mathrm{2020}\right)} \left({x}\right)\:=\: \\ $$

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