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Question Number 44730 Answers: 1 Comments: 1

Question Number 44729 Answers: 4 Comments: 0

Question Number 44716 Answers: 1 Comments: 0

prove: 1 + 11 + 111 + .... + ((111 ...111)/(n times)) = ((10^(n + 1) − 9n − 10)/(81))

$$\mathrm{prove}:\:\:\:\:\:\mathrm{1}\:+\:\mathrm{11}\:+\:\mathrm{111}\:+\:....\:+\:\frac{\mathrm{111}\:...\mathrm{111}}{\mathrm{n}\:\mathrm{times}}\:\:=\:\:\frac{\mathrm{10}^{\mathrm{n}\:+\:\mathrm{1}} \:−\:\mathrm{9n}\:−\:\mathrm{10}}{\mathrm{81}} \\ $$

Question Number 44712 Answers: 3 Comments: 0

Question Number 44708 Answers: 1 Comments: 0

Let A,B be two n×n matrices such that A+B=AB then prove : AB=BA ?

$${Let}\:{A},{B}\:{be}\:{two}\:{n}×{n}\:{matrices}\:{such} \\ $$$${that}\:{A}+{B}={AB}\:{then}\:{prove}\:: \\ $$$${AB}={BA}\:? \\ $$

Question Number 44702 Answers: 0 Comments: 0

Question Number 44697 Answers: 0 Comments: 0

prove that:− ∫_0 ^∞ (t^(a−1) /(1+t))dt = (𝛑/(sin(𝛑a)))

$$\boldsymbol{{prove}}\:\boldsymbol{{that}}:− \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{t}}^{\boldsymbol{{a}}−\mathrm{1}} }{\mathrm{1}+\boldsymbol{{t}}}\boldsymbol{{dt}}\:=\:\frac{\boldsymbol{\pi}}{\boldsymbol{{sin}}\left(\boldsymbol{\pi{a}}\right)} \\ $$

Question Number 44704 Answers: 2 Comments: 0

Find the general solution of : 311x − 112y = 73

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\::\:\:\:\:\:\:\:\:\:\:\:\mathrm{311x}\:−\:\mathrm{112y}\:=\:\mathrm{73} \\ $$

Question Number 44706 Answers: 0 Comments: 4

let f_α (x) = ((cos(αx))/(1+x^2 )) 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie 3) give ∫_0 ^x f_α (t) dt at form of serie 4) developp ∫_0 ^∞ f_α (t)dt at integr serie .

$${let}\:{f}_{\alpha} \left({x}\right)\:=\:\frac{{cos}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{give}\:\int_{\mathrm{0}} ^{{x}} \:{f}_{\alpha} \left({t}\right)\:{dt}\:\:{at}\:{form}\:{of}\:{serie}\: \\ $$$$\left.\mathrm{4}\right)\:{developp}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:{f}_{\alpha} \left({t}\right){dt}\:\:{at}\:\:{integr}\:{serie}\:. \\ $$

Question Number 44696 Answers: 1 Comments: 1

∫(1/(1+x^4 ))dx = ?

$$\int\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{4}} }\boldsymbol{\mathrm{dx}}\:=\:? \\ $$

Question Number 44695 Answers: 0 Comments: 2

∫(e^(√(t−1)) /t)dt = ?

$$\int\frac{\boldsymbol{\mathrm{e}}^{\sqrt{\boldsymbol{\mathrm{t}}−\mathrm{1}}} }{\boldsymbol{\mathrm{t}}}\boldsymbol{\mathrm{dt}}\:=\:? \\ $$

Question Number 44691 Answers: 1 Comments: 1

Question Number 44676 Answers: 1 Comments: 6

Question Number 44652 Answers: 2 Comments: 4

Prove that lim_(x→0) (((1+ax)^(1/b) −1)/x) = (a/b).

$${Prove}\:{that}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+{ax}\right)^{\frac{\mathrm{1}}{{b}}} −\mathrm{1}}{{x}}\:=\:\frac{{a}}{{b}}. \\ $$

Question Number 44639 Answers: 0 Comments: 1

∫(1/(1+(log x)^2 ))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{1}+\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=? \\ $$$$ \\ $$

Question Number 44636 Answers: 1 Comments: 0

Question Number 44654 Answers: 1 Comments: 4

∫(e^x /(1+x^2 ))dx=?

$$\int\frac{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} }{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 44623 Answers: 1 Comments: 0

given that sin^(−1) x+sin^(−1) y=c show that (dy/dx)+(√((1−y^2 )/(1−x^2 )))=0

$$\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{sin}}^{−\mathrm{1}} \boldsymbol{{x}}+\boldsymbol{\mathrm{sin}}^{−\mathrm{1}} \boldsymbol{{y}}=\boldsymbol{\mathrm{c}} \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}+\sqrt{\frac{\mathrm{1}−\boldsymbol{{y}}^{\mathrm{2}} }{\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} }}=\mathrm{0} \\ $$

Question Number 44622 Answers: 1 Comments: 0

if y=ln[tan((𝛑/4)+(x/2))] show that (dy/dx)=secx

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{{y}}=\boldsymbol{\mathrm{ln}}\left[\boldsymbol{\mathrm{tan}}\left(\frac{\boldsymbol{\pi}}{\mathrm{4}}+\frac{\boldsymbol{{x}}}{\mathrm{2}}\right)\right]\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}=\boldsymbol{\mathrm{sec}{x}} \\ $$

Question Number 44621 Answers: 2 Comments: 0

Question Number 44613 Answers: 0 Comments: 1

Question Number 44612 Answers: 2 Comments: 0

Prove that One factor of determinant (((a^2 +x),( ab),( ac)),(( ab),(b^2 +x),( cb)),(( ca),( cb),(c^2 +x))) is x^2 .

$${Prove}\:{that}\:\mathrm{One}\:\mathrm{factor}\:\mathrm{of}\begin{vmatrix}{{a}^{\mathrm{2}} +{x}}&{\:\:{ab}}&{\:\:{ac}}\\{\:\:{ab}}&{{b}^{\mathrm{2}} +{x}}&{\:\:{cb}}\\{\:\:{ca}}&{\:\:{cb}}&{{c}^{\mathrm{2}} +{x}}\end{vmatrix}\:\mathrm{is}\:{x}^{\mathrm{2}} . \\ $$

Question Number 44604 Answers: 1 Comments: 1

∫[((log x − 1)/(1+(log x)^2 ))]^2 dx = (x/((log x)^2 +1))+C

$$\int\left[\frac{\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\:\:−\:\:\mathrm{1}}{\mathrm{1}+\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} }\right]^{\mathrm{2}} \boldsymbol{\mathrm{dx}}\:\:=\:\:\frac{\boldsymbol{\mathrm{x}}}{\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} +\mathrm{1}}+\boldsymbol{\mathrm{C}} \\ $$

Question Number 44602 Answers: 0 Comments: 0

∫(e^x /(1+x^2 )) dx = ?

$$\int\frac{\boldsymbol{{e}}^{\boldsymbol{\mathrm{x}}} }{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}}\:=\:\:? \\ $$

Question Number 44587 Answers: 1 Comments: 2

calculate ∫_0 ^∞ (dt/(1+t^(2018) ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2018}} } \\ $$

Question Number 44584 Answers: 1 Comments: 5

Prove that if a, b, c ∈ Z and a^2 + b^2 = c^2 , then 3 ∣ ab

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\:{a},\:{b},\:{c}\:\in\:\mathbb{Z}\:\:\mathrm{and}\:\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:{c}^{\mathrm{2}} ,\:\mathrm{then} \\ $$$$\mathrm{3}\:\mid\:{ab} \\ $$

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