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AllQuestion and Answers: Page 1238

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) =

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

Question Number 90629 Answers: 1 Comments: 1

If sin^(−1) (1−x)−2 sin^(−1) x = (π/2), then x=

$$\mathrm{If}\:\:\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)−\mathrm{2}\:\mathrm{sin}^{−\mathrm{1}} {x}\:=\:\frac{\pi}{\mathrm{2}},\:\mathrm{then}\:{x}= \\ $$

Question Number 90628 Answers: 0 Comments: 1

The value of sin (π/(14)) sin ((3π)/(14)) sin ((5π)/(14)) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{sin}\:\frac{\pi}{\mathrm{14}}\:\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{14}}\:\mathrm{sin}\:\frac{\mathrm{5}\pi}{\mathrm{14}}\:\:\mathrm{is} \\ $$

Question Number 90625 Answers: 0 Comments: 0

∫e^(arcsinx) dx

$$\int{e}^{{arcsinx}} {dx} \\ $$

Question Number 90609 Answers: 0 Comments: 3

find the infinite sumΣ_(n=0) ^∞ (F_n /2^n ) where F_n =(1/(√5))(((1+(√5))/2))^(n+1) −(1/(√5))(((1−(√5))/2))^(n+1)

$${find}\:{the}\:{infinite}\:{sum}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{F}_{{n}} }{\mathrm{2}^{{n}} }\: \\ $$$${where}\:{F}_{{n}} =\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}+\mathrm{1}} −\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}+\mathrm{1}} \\ $$

Question Number 90596 Answers: 1 Comments: 0

If sin(28) = a and cos(32) = b Find (i) cos(28) (ii) cos(64) (iii) sin(4)

$$\mathrm{If}\:\:\:\mathrm{sin}\left(\mathrm{28}\right)\:\:=\:\:\mathrm{a}\:\:\:\:\mathrm{and}\:\:\:\mathrm{cos}\left(\mathrm{32}\right)\:\:=\:\:\mathrm{b} \\ $$$$\mathrm{Find}\:\:\left(\mathrm{i}\right)\:\:\mathrm{cos}\left(\mathrm{28}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{ii}\right)\:\mathrm{cos}\left(\mathrm{64}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{iii}\right)\:\mathrm{sin}\left(\mathrm{4}\right) \\ $$

Question Number 90594 Answers: 0 Comments: 3

Question Number 90592 Answers: 0 Comments: 3

∫_0 ^(√(arccos(((−2φ)/π)+1))) x sin(x^2 ) dx

$$\int_{\mathrm{0}} ^{\sqrt{{arccos}\left(\frac{−\mathrm{2}\phi}{\pi}+\mathrm{1}\right)}} {x}\:{sin}\left({x}^{\mathrm{2}} \right)\:{dx} \\ $$

Question Number 90590 Answers: 1 Comments: 2

Question Number 90589 Answers: 0 Comments: 3

∫(1/(x+(√(x^2 +x+1))))dx

$$\int\frac{\mathrm{1}}{{x}+\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$

Question Number 90588 Answers: 1 Comments: 0

∫(√(x+(√(x^2 +6 ))))dx

$$\int\sqrt{{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{6}\:\:}}{dx} \\ $$

Question Number 90575 Answers: 0 Comments: 0

Solve the differential equation: x y_3 +(1−2x^2 )y_2 −8x _ y_1 −4y= e^x

$$\:\boldsymbol{\mathrm{Solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}}: \\ $$$$\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{y}}_{\mathrm{3}} +\left(\mathrm{1}−\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\boldsymbol{\mathrm{y}}_{\mathrm{2}} −\mathrm{8}\boldsymbol{\mathrm{x}}\:_{\:} \boldsymbol{\mathrm{y}}_{\mathrm{1}} −\mathrm{4}\boldsymbol{\mathrm{y}}=\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} \\ $$

Question Number 90574 Answers: 0 Comments: 2

Use gamma function to prove (i) . ∫_0 ^( (𝛑/8)) cos^3 4x dx= (1/6). (ii). ∫_0 ^( (𝛑/6)) cos^4 3𝛉 sin^2 6𝛉 d𝛉 = ((5𝛑)/(192)).

Question Number 90570 Answers: 1 Comments: 1

find the sum of Σ_(n=1) ^∞ (((−1)^n )/((2n+1)3^n ))

$${find}\:{the}\:{sum}\:{of}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\mathrm{3}^{{n}} } \\ $$

Question Number 90566 Answers: 0 Comments: 0

∫_0 ^(infinity) Sin^4 3x/x^2 dx

$$\int_{\mathrm{0}} ^{\mathrm{infinity}} \mathrm{Sin}^{\mathrm{4}} \mathrm{3x}/\mathrm{x}^{\mathrm{2}} \mathrm{dx} \\ $$

Question Number 90564 Answers: 0 Comments: 1

find lim_(x→0) (((^3 (√(1+cos(2x)))−(^3 (√2)))/(x^2 sin(3x)))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\left(^{\mathrm{3}} \sqrt{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)}−\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)\right.}{{x}^{\mathrm{2}} {sin}\left(\mathrm{3}{x}\right)} \\ $$

Question Number 90562 Answers: 0 Comments: 2

prove that Σ_(n=1) ^∞ (1/(2n(2n−1)))=ln2

$${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right)}={ln}\mathrm{2} \\ $$

Question Number 90561 Answers: 0 Comments: 2

prove that Π_(n=2) ^∞ (1−(1/n^2 ))=(1/2)

$${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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