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Question Number 108582 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((ln(1+x))/(1+x^2 )) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 108573 Answers: 3 Comments: 0

please: in AB^Δ C prove that: ((cos(A))/(sin(B)sin(C))) +((cos(B))/(sin(A)sin(C)))+((cos(C))/(sin(A)sin(B))) =2

$$\:\:\:\:\:\:\:\:\:\mathrm{please}:\:\:\:\mathrm{in}\:\mathrm{A}\overset{\Delta} {\mathrm{B}C}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\:\frac{{cos}\left(\mathrm{A}\right)}{{sin}\left(\mathrm{B}\right){sin}\left(\mathrm{C}\right)}\:+\frac{{cos}\left(\mathrm{B}\right)}{{sin}\left(\mathrm{A}\right){sin}\left(\mathrm{C}\right)}+\frac{{cos}\left(\mathrm{C}\right)}{{sin}\left(\mathrm{A}\right){sin}\left(\mathrm{B}\right)}\:=\mathrm{2}\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 108571 Answers: 2 Comments: 0

Question Number 108570 Answers: 0 Comments: 0

Question Number 108568 Answers: 0 Comments: 0

Question Number 108553 Answers: 0 Comments: 0

If a_1 , a_2 , a_3 , be an AP, then prove that: Σ_(n = 1) ^(2m) (− 1)^(n − 1) a_n ^2 = (m/(2m − 1))(a_n ^2 − a_(2m) ^2 )

$$\mathrm{If}\:\:\:\:\mathrm{a}_{\mathrm{1}} ,\:\:\mathrm{a}_{\mathrm{2}} ,\:\:\mathrm{a}_{\mathrm{3}} ,\:\:\:\:\mathrm{be}\:\mathrm{an}\:\mathrm{AP},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\:\:\:\underset{\mathrm{n}\:\:=\:\:\mathrm{1}} {\overset{\mathrm{2m}} {\sum}}\:\left(−\:\mathrm{1}\right)^{\mathrm{n}\:\:−\:\:\mathrm{1}} \:\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:\:=\:\:\:\frac{\mathrm{m}}{\mathrm{2m}\:\:−\:\:\mathrm{1}}\left(\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:−\:\:\mathrm{a}_{\mathrm{2m}} ^{\mathrm{2}} \right) \\ $$

Question Number 108538 Answers: 0 Comments: 1

Question Number 108548 Answers: 0 Comments: 0

Question Number 108547 Answers: 1 Comments: 0

Solve the following equation: cosz =2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation}: \\ $$$$\mathrm{cosz}\:=\mathrm{2} \\ $$

Question Number 108507 Answers: 1 Comments: 0

∫_0 ^(π/6) (√((3cos2x−1)/(cos^2 (x)))) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \sqrt{\frac{\mathrm{3}{cos}\mathrm{2}{x}−\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)}}\:{dx} \\ $$

Question Number 108506 Answers: 1 Comments: 0

Question Number 108502 Answers: 1 Comments: 1

Question Number 108498 Answers: 2 Comments: 0

Question Number 108491 Answers: 1 Comments: 0

(1+2x)y′′+(4x−2)y′−8y=0

$$\left(\mathrm{1}+\mathrm{2x}\right)\mathrm{y}''+\left(\mathrm{4x}−\mathrm{2}\right)\mathrm{y}'−\mathrm{8y}=\mathrm{0} \\ $$

Question Number 108483 Answers: 0 Comments: 3

Question Number 108480 Answers: 6 Comments: 0

Question Number 108516 Answers: 2 Comments: 7

If x+(1/x)=2(x≠0), prove that x^n +(1/x^n )=2 ∀ n∈ Z

$$\mathrm{If}\:\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}=\mathrm{2}\left(\mathrm{x}\neq\mathrm{0}\right),\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\mathrm{x}^{\mathrm{n}} +\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{n}} }=\mathrm{2}\:\:\forall\:{n}\in\:\mathbb{Z} \\ $$

Question Number 108476 Answers: 2 Comments: 0

S_n =Σ_(k=1 ) ^n k^2 (−1)^k C_n ^k =? please help

$$\boldsymbol{{S}}_{{n}} =\underset{{k}=\mathrm{1}\:} {\overset{{n}} {\sum}}{k}^{\mathrm{2}} \left(−\mathrm{1}\right)^{{k}} \boldsymbol{{C}}_{\boldsymbol{{n}}} ^{\boldsymbol{{k}}} =? \\ $$$$\:\:\boldsymbol{{ple}}{ase}\:{help} \\ $$

Question Number 108475 Answers: 2 Comments: 0

(x^2 +1)y^′ +2xy=x(√(x^2 +1)) please solve this differential equation

$$\left({x}^{\mathrm{2}} +\mathrm{1}\right)\boldsymbol{{y}}^{'} +\mathrm{2}{x}\boldsymbol{{y}}={x}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$${please}\:{solve}\:{this}\:{differential} \\ $$$${equation} \\ $$

Question Number 108474 Answers: 1 Comments: 0

U_n =Σ_(k=0) ^(2n−1) (1/(2n+k))=? lim_(n>∞) U_n =?

$$\boldsymbol{{U}}_{\boldsymbol{{n}}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{\mathrm{2}{n}+{k}}=? \\ $$$$\boldsymbol{{li}}\underset{\boldsymbol{{n}}>\infty} {\boldsymbol{{m}U}}_{{n}} =? \\ $$

Question Number 108473 Answers: 2 Comments: 0

3x^3 +4x−3=0 solve this problem.

$$\mathrm{3x}^{\mathrm{3}} +\mathrm{4x}−\mathrm{3}=\mathrm{0} \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{problem}. \\ $$

Question Number 108469 Answers: 2 Comments: 0

((⊂BeMath⊃)/∩) (1)lim_(x→0) ((1−cos x (√(cos 2x)) (√(cos 3x))...(√(cos nx)))/x^2 ) ? (2)x^2 y′′+xy′−4y=0; y(1)=2 and y′(1)=0 (3)find the probability that a person throwing three coins at once will get all the face or everything back for second time at 5 the throws.

$$\:\:\:\frac{\subset\mathcal{B}{e}\mathcal{M}{ath}\supset}{\cap} \\ $$$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:\sqrt{\mathrm{cos}\:\mathrm{3}{x}}...\sqrt{\mathrm{cos}\:{nx}}}{{x}^{\mathrm{2}} }\:? \\ $$$$\left(\mathrm{2}\right){x}^{\mathrm{2}} {y}''+{xy}'−\mathrm{4}{y}=\mathrm{0};\:{y}\left(\mathrm{1}\right)=\mathrm{2}\:{and} \\ $$$$\:\:\:{y}'\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\left(\mathrm{3}\right){find}\:{the}\:{probability}\:{that}\:{a}\:{person}\:\:{throwing}\:{three} \\ $$$${coins}\:{at}\:{once}\:{will}\:{get}\:{all}\:{the}\:{face}\:{or}\: \\ $$$${everything}\:{back}\:{for}\:{second}\:{time}\:{at} \\ $$$$\mathrm{5}\:{the}\:{throws}. \\ $$

Question Number 108466 Answers: 0 Comments: 1

Question Number 108463 Answers: 0 Comments: 0

arcsin(sin10)

$${arcsin}\left({sin}\mathrm{10}\right) \\ $$

Question Number 108461 Answers: 1 Comments: 1

Question Number 108456 Answers: 3 Comments: 1

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