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AllQuestion and Answers: Page 138

Question Number 208322 Answers: 1 Comments: 0

Question Number 208318 Answers: 1 Comments: 0

calcul lim n→+∞ ∫_0 ^(+∞) ((cos(nx))/((nx+1)(1+x^2 ) ))dx

$${calcul}\:\:\:{lim}\:{n}\rightarrow+\infty \\ $$$$\int_{\mathrm{0}} ^{+\infty} \:\frac{{cos}\left({nx}\right)}{\left({nx}+\mathrm{1}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:}{dx} \\ $$

Question Number 208316 Answers: 1 Comments: 0

∫ ((x^2 + 3)/(x^2 (x + 1)(x^2 + 1)^2 )) dx

$$\int\:\frac{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{3}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}\:\:+\:\:\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 208312 Answers: 1 Comments: 0

lim_(x→0 ((a^x −1)/x) = log a)

$${lim}_{{x}\rightarrow\mathrm{0}\:\frac{{a}^{{x}} −\mathrm{1}}{{x}}\:=\:{log}\:{a}} \\ $$

Question Number 208306 Answers: 2 Comments: 1

calcul / lim n→+∞ ∫_0 ^(+∞) f_n (x) f_n (x)= arctan((x/n))e^(−x) dx

$${calcul}\:/\:{lim}\:{n}\rightarrow+\infty\:\int_{\mathrm{0}} ^{+\infty} \:{f}_{{n}} \left({x}\right) \\ $$$$\:{f}_{{n}} \left({x}\right)=\:{arctan}\left(\frac{{x}}{{n}}\right){e}^{−{x}} {dx} \\ $$

Question Number 208303 Answers: 1 Comments: 0

Resolver (∂^2 u/∂y^2 ) − x^2 u = xe^(4y)

$${Resolver} \\ $$$$\frac{\partial^{\mathrm{2}} {u}}{\partial{y}^{\mathrm{2}} }\:−\:{x}^{\mathrm{2}} {u}\:=\:{xe}^{\mathrm{4}{y}} \\ $$

Question Number 208293 Answers: 1 Comments: 0

( / ) + ( ^2 / ^2 ) + ( ^2 / ^3 ) + ( ^2 / ^4 ) + ( ^2 / ^5 ) + ...

$$\:\:\:\:\: \frac{ }{ }\:+\:\frac{ ^{\mathrm{2}} }{ ^{\mathrm{2}} }\:+\:\frac{ ^{\mathrm{2}} }{ ^{\mathrm{3}} }\:+\:\frac{ ^{\mathrm{2}} }{ ^{\mathrm{4}} }\:+\:\frac{ ^{\mathrm{2}} }{ ^{\mathrm{5}} }\:+\:...\: \\ $$$$\:\:\:\:\: \\ $$

Question Number 208292 Answers: 1 Comments: 0

let T be a n×n matrix with integral entries and Q = T + (1/2)I where I denote the n×n identity matrix then prove that matrix Q is invertible

$$\:\mathrm{let}\:\mathrm{T}\:\mathrm{be}\:\mathrm{a}\:{n}×{n}\:\mathrm{matrix}\:\mathrm{with}\:\mathrm{integral}\: \\ $$$$\:\mathrm{entries}\:\mathrm{and}\:\:\mathrm{Q}\:=\:\mathrm{T}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{I}\:\:\:\mathrm{where}\:\mathrm{I}\:\mathrm{denote} \\ $$$$\:\:\mathrm{the}\:\mathrm{n}×\mathrm{n}\:\mathrm{identity}\:\mathrm{matrix}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\:\:\mathrm{that}\:\mathrm{matrix}\:\mathrm{Q}\:\mathrm{is}\:\mathrm{invertible} \\ $$

Question Number 208288 Answers: 2 Comments: 0

Question Number 208282 Answers: 1 Comments: 2

If cosα−cosβ = (1/5) sinα + sinβ = (1/2) Find cos(α + β) = ?

$$\mathrm{If}\:\:\:\mathrm{cos}\alpha−\mathrm{cos}\beta\:=\:\frac{\mathrm{1}}{\mathrm{5}}\:\mathrm{sin}\alpha\:+\:\mathrm{sin}\beta\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{Find}\:\:\:\mathrm{cos}\left(\alpha\:+\:\beta\right)\:=\:? \\ $$

Question Number 208280 Answers: 1 Comments: 0

L=∫_0 ^(4/π) ln(cosx)dx

$${L}=\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left({cosx}\right){dx} \\ $$

Question Number 208277 Answers: 2 Comments: 0

Question Number 208269 Answers: 2 Comments: 1

Question Number 208264 Answers: 1 Comments: 0

e^x .

$$\:\: \mathrm{e}^{\mathrm{x}} . \\ $$

Question Number 208263 Answers: 1 Comments: 0

Question Number 208259 Answers: 2 Comments: 0

Question Number 208256 Answers: 0 Comments: 1

^2 ((( π)/ )) + ^2 ((( π)/ )) + ^2 ((( π)/ ))=?

$$\: ^{\mathrm{2}} \left(\frac{ \pi}{ }\right)\:+\: ^{\mathrm{2}} \left(\frac{ \pi}{ }\right)\:+\: ^{\mathrm{2}} \left(\frac{ \pi}{ }\right)=? \\ $$

Question Number 208252 Answers: 1 Comments: 0

cos (((2π)/(21)))cos (((4π)/(21)))cos (((8π)/(21)))cos (((10π)/(22)))cos (((16π)/(21)))cos (((20π)/(21)))=?

$$\:\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{10}\pi}{\mathrm{22}}\right)\mathrm{cos}\:\left(\frac{\mathrm{16}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{20}\pi}{\mathrm{21}}\right)=? \\ $$

Question Number 208251 Answers: 0 Comments: 0

Question Number 208245 Answers: 1 Comments: 0

K=∫_0 ^(4/π) ln(cosx)dx

$${K}=\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left({cosx}\right){dx} \\ $$

Question Number 208242 Answers: 1 Comments: 1

Question Number 208241 Answers: 1 Comments: 0

Solve for p, q, r p+q+r=α p^2 +q^2 +r^2 =β pq=r

$$\mathrm{Solve}\:\mathrm{for}\:{p},\:{q},\:{r} \\ $$$${p}+{q}+{r}=\alpha \\ $$$${p}^{\mathrm{2}} +{q}^{\mathrm{2}} +{r}^{\mathrm{2}} =\beta \\ $$$${pq}={r} \\ $$

Question Number 208238 Answers: 1 Comments: 0

Show that (π/4) < ∫_0 ^1 (√(1−x^4 ))dx using x = sint show that ∫_0 ^1 (√(1−x^4 ))dx<((2(√2))/3) using (∫_0 ^1 f(x)g(x)dx)^2 <∫_0 ^1 (f(x))^2 dx∫_0 ^1 (g(x))^2 dx

$$\mathrm{S}{how}\:{that} \\ $$$$\frac{\pi}{\mathrm{4}}\:<\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}\:{using}\:{x}\:=\:{sint} \\ $$$${show}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx}<\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{3}} \\ $$$${using}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){g}\left({x}\right){dx}\right)^{\mathrm{2}} <\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}\left({x}\right)\right)^{\mathrm{2}} {dx}\int_{\mathrm{0}} ^{\mathrm{1}} \left({g}\left({x}\right)\right)^{\mathrm{2}} {dx} \\ $$

Question Number 208235 Answers: 2 Comments: 0

Question Number 208218 Answers: 1 Comments: 4

a_n numbers series If S_(16) − S_(13) = S_(106) − S_(103) Find: ((3a_3 + 4a_4 + 5a_5 )/(2a_(12) )) = ?

$$\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:\:\mathrm{numbers}\:\mathrm{series} \\ $$$$\mathrm{If}\:\:\mathrm{S}_{\mathrm{16}} \:−\:\mathrm{S}_{\mathrm{13}} \:\:=\:\:\mathrm{S}_{\mathrm{106}} \:−\:\mathrm{S}_{\mathrm{103}} \\ $$$$\mathrm{Find}:\:\:\:\:\frac{\mathrm{3a}_{\mathrm{3}} \:+\:\mathrm{4a}_{\mathrm{4}} \:+\:\mathrm{5a}_{\mathrm{5}} }{\mathrm{2a}_{\mathrm{12}} }\:\:=\:\:? \\ $$

Question Number 208217 Answers: 3 Comments: 0

1^2 +2^2 +3^2 +5^2 +8^2 +13^2 +21^2 =?

$$\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{8}^{\mathrm{2}} +\mathrm{13}^{\mathrm{2}} +\mathrm{21}^{\mathrm{2}} =? \\ $$

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