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Question Number 221770 Answers: 0 Comments: 0

Prove:∫_0 ^1 ((arcsinx)/(1+x^4 ))dx=(((√2)π^2 )/(16))−(((√2)π)/8)ln((√2)−1)+2Σ_(n=0) ^∞ (((−)^n )/((2n+1)))∣z_0 ∣^(2n+1) sin((π/4)−(2n+1)β)

$$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{arcsin}{x}}{\mathrm{1}+{x}^{\mathrm{4}} }{dx}=\frac{\sqrt{\mathrm{2}}\pi^{\mathrm{2}} }{\mathrm{16}}−\frac{\sqrt{\mathrm{2}}\pi}{\mathrm{8}}\mathrm{ln}\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)+\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)}\mid{z}_{\mathrm{0}} \mid^{\mathrm{2}{n}+\mathrm{1}} \mathrm{sin}\left(\frac{\pi}{\mathrm{4}}−\left(\mathrm{2}{n}+\mathrm{1}\right)\beta\right) \\ $$

Question Number 221769 Answers: 0 Comments: 0

∫_0 ^π (a/(a−cos^(2n) x))dx=?,a>1 ∫_0 ^π (2/(2−cos^4 x))dx=? lim_(m→∞) ∫_0 ^π ((cos^(2n) (2mx))/(a−cos^(2n) x))dx=?,a>1,n∈N^+

$$\int_{\mathrm{0}} ^{\pi} \frac{{a}}{{a}−\mathrm{cos}^{\mathrm{2}{n}} {x}}{dx}=?,{a}>\mathrm{1} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{2}}{\mathrm{2}−\mathrm{cos}^{\mathrm{4}} {x}}{dx}=? \\ $$$$\underset{{m}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{cos}^{\mathrm{2}{n}} \left(\mathrm{2}{mx}\right)}{{a}−\mathrm{cos}^{\mathrm{2}{n}} {x}}{dx}=?,{a}>\mathrm{1},{n}\in\mathbb{N}^{+} \\ $$

Question Number 221760 Answers: 1 Comments: 0

Question Number 221754 Answers: 1 Comments: 0

((81))^(1/((64))^(1/((27))^(1/( 3^4^0^4^3 )) ) ) )^((√4))

$$\left.\sqrt[{\sqrt[{\sqrt[{\:\mathrm{3}^{\mathrm{4}^{\mathrm{0}^{\mathrm{4}^{\mathrm{3}} } } } }]{\mathrm{27}}}]{\mathrm{64}}}]{\mathrm{81}}\right)^{\sqrt{\mathrm{4}}} \\ $$

Question Number 221733 Answers: 0 Comments: 19

Question Number 221721 Answers: 2 Comments: 0

Question Number 221707 Answers: 1 Comments: 13

Question Number 221697 Answers: 2 Comments: 0

Is (√i) an imaginary number (i=(√(−1))) answer with logic

$${Is}\:\sqrt{{i}}\:{an}\:{imaginary}\:{number}\:\left({i}=\sqrt{−\mathrm{1}}\right)\:{answer}\:{with}\:{logic} \\ $$

Question Number 221686 Answers: 3 Comments: 2

Question Number 221668 Answers: 0 Comments: 0

Question Number 221663 Answers: 0 Comments: 3

Prove:∫_0 ^1 Π_(k=1) ^∞ (1−x^k )dx=((4π(√3))/( (√(23))))∙((sinh(((√(23))π)/6))/(2 cosh(((√(23))π)/3)−1))

$$\mathrm{Prove}:\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\mathrm{1}−{x}^{{k}} \right){dx}=\frac{\mathrm{4}\pi\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{23}}}\centerdot\frac{\mathrm{sinh}\frac{\sqrt{\mathrm{23}}\pi}{\mathrm{6}}}{\mathrm{2}\:\mathrm{cosh}\frac{\sqrt{\mathrm{23}}\pi}{\mathrm{3}}−\mathrm{1}} \\ $$

Question Number 221661 Answers: 1 Comments: 1

Question Number 221647 Answers: 0 Comments: 1

solve for x. x^1 + x^2 + x^3 = 4096

$${solve}\:{for}\:{x}. \\ $$$${x}^{\mathrm{1}} \:+\:{x}^{\mathrm{2}} \:+\:{x}^{\mathrm{3}} \:\:=\:\:\mathrm{4096} \\ $$

Question Number 221638 Answers: 1 Comments: 0

Question Number 221637 Answers: 1 Comments: 0

Question Number 221626 Answers: 3 Comments: 0

Question Number 221620 Answers: 1 Comments: 0

Solve for x ((7x))^(1/3) =(√x)[x≠0]

$${Solve}\:{for}\:{x} \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{7}{x}}=\sqrt{{x}}\left[{x}\neq\mathrm{0}\right] \\ $$

Question Number 221618 Answers: 3 Comments: 0

solve for x 2^x +4^x =8^x

$${solve}\:{for}\:{x} \\ $$$$\mathrm{2}^{{x}} +\mathrm{4}^{{x}} =\mathrm{8}^{{x}} \\ $$

Question Number 221669 Answers: 2 Comments: 3

Question Number 221601 Answers: 2 Comments: 0

∫ ((sin 2x)/(1 + sin 3x)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\frac{{sin}\:\mathrm{2}{x}}{\mathrm{1}\:+\:{sin}\:\mathrm{3}{x}}\:{dx} \\ $$$$ \\ $$

Question Number 221592 Answers: 2 Comments: 0

Question Number 221588 Answers: 1 Comments: 0

∫ ((8t − 8t^( 3) )/(t^( 6) + 6t^5 + 3t^( 4) − 20t^3 + 3t^2 + 6t + 1)) dt

$$ \\ $$$$\:\:\:\:\int\:\frac{\mathrm{8}{t}\:−\:\mathrm{8}{t}^{\:\mathrm{3}} }{{t}^{\:\mathrm{6}} \:+\:\mathrm{6}{t}^{\mathrm{5}} \:+\:\mathrm{3}{t}^{\:\mathrm{4}} \:−\:\mathrm{20}{t}^{\mathrm{3}} \:+\:\mathrm{3}{t}^{\mathrm{2}} \:+\:\mathrm{6}{t}\:+\:\mathrm{1}}\:{dt}\:\:\:\: \\ $$$$ \\ $$

Question Number 221587 Answers: 1 Comments: 0

∫_( 2) ^( 3) ((tan^(− 1) (x))/(1 − x^2 )) dx

$$\int_{\:\mathrm{2}} ^{\:\mathrm{3}} \:\frac{\mathrm{tan}^{−\:\mathrm{1}} \left(\mathrm{x}\right)}{\mathrm{1}\:\:−\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 221586 Answers: 1 Comments: 0

∫ ((sin 2x)/(1 + 3x )) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{1}\:+\:\mathrm{3}{x}\:}\:{dx} \\ $$$$ \\ $$

Question Number 221585 Answers: 6 Comments: 1

solve for x ∈R (x^3 −6)^3 =x+6

$${solve}\:{for}\:{x}\:\in{R} \\ $$$$\left({x}^{\mathrm{3}} −\mathrm{6}\right)^{\mathrm{3}} ={x}+\mathrm{6} \\ $$

Question Number 221583 Answers: 1 Comments: 0

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