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Question Number 111082    Answers: 1   Comments: 1

[∫_0 ^∞ JS dx ] ∫_0 ^(π/2) ((sin (x)(4+sin^2 (x)))/((4−sin^2 (x))^2 )) dx ?

$$\:\:\:\left[\int_{\mathrm{0}} ^{\infty} {JS}\:{dx}\:\right] \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{sin}\:\left({x}\right)\left(\mathrm{4}+\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)\right)}{\left(\mathrm{4}−\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)\right)^{\mathrm{2}} }\:{dx}\:? \\ $$

Question Number 111080    Answers: 1   Comments: 0

(√(bemath)) (1)Σ_(k=50) ^(100) (1/(k(151−k))) ? (2) without L′Hopital and series find the value of lim_(x→0) ((xcos x−sin x)/(x^2 sin x))

$$\:\:\sqrt{\mathrm{bemath}} \\ $$$$\left(\mathrm{1}\right)\underset{\mathrm{k}=\mathrm{50}} {\overset{\mathrm{100}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{k}\left(\mathrm{151}−\mathrm{k}\right)}\:? \\ $$$$\left(\mathrm{2}\right)\:\mathrm{without}\:\mathrm{L}'\mathrm{Hopital}\:\mathrm{and}\:\mathrm{series}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{xcos}\:\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} \mathrm{sin}\:\mathrm{x}} \\ $$

Question Number 111079    Answers: 1   Comments: 0

Question Number 111027    Answers: 0   Comments: 0

★((log _(JS) (farmer))/)★ (1)∫ ((tan (ln x)tan (ln ((x/2)))dx)/x) (2) sin (cos x) < cos (sin x) ; where 0≤x≤2π

$$\:\:\bigstar\frac{\mathrm{log}\:_{{JS}} \left({farmer}\right)}{}\bigstar \\ $$$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{tan}\:\left(\mathrm{ln}\:{x}\right)\mathrm{tan}\:\left(\mathrm{ln}\:\left(\frac{{x}}{\mathrm{2}}\right)\right){dx}}{{x}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{sin}\:\left(\mathrm{cos}\:{x}\right)\:<\:\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)\:;\:{where} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\pi \\ $$

Question Number 111025    Answers: 1   Comments: 0

calculate ∫_0 ^∞ ((x^2 ln(x))/((1+x)^4 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{ln}\left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 111024    Answers: 1   Comments: 0

calculate ∫_0 ^∞ ((x^2 lnx)/((1+x^2 )^3 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{lnx}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\mathrm{dx} \\ $$

Question Number 111023    Answers: 1   Comments: 1

∫((sin(x))/(1+x^2 ))dx

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

Question Number 111011    Answers: 1   Comments: 0

solve: y^(′′) +y^′ =tanx

$${solve}:\:{y}^{''} +{y}^{'} ={tanx} \\ $$

Question Number 111010    Answers: 1   Comments: 0

∫e^x tanx dx

$$\int{e}^{{x}} \:{tanx}\:{dx} \\ $$

Question Number 111008    Answers: 0   Comments: 3

((√(x+1))/(y+2)) + ((√(y+2))/(x+1)) =1 => x=?

$$\frac{\sqrt{\boldsymbol{{x}}+\mathrm{1}}}{\boldsymbol{{y}}+\mathrm{2}}\:+\:\frac{\sqrt{\boldsymbol{{y}}+\mathrm{2}}}{\boldsymbol{{x}}+\mathrm{1}}\:=\mathrm{1}\:\:\:\:\:\:=>\:\:\boldsymbol{{x}}=? \\ $$

Question Number 111006    Answers: 0   Comments: 2

The vectors p,q and r are mutially perpendicularwith ∣q∣=3 and ∣r∣=(√(5.4 )) .If X= 7p+5q+7r and Y=2p+3q−5r are perpendicular, find∣p∣.

$$\mathrm{The}\:\mathrm{vectors}\:\boldsymbol{\mathrm{p}},\boldsymbol{\mathrm{q}}\:\mathrm{and}\:\boldsymbol{\mathrm{r}}\:\mathrm{are}\:\mathrm{mutially}\:\mathrm{perpendicularwith} \\ $$$$\mid\boldsymbol{\mathrm{q}}\mid=\mathrm{3}\:\mathrm{and}\:\mid\boldsymbol{\mathrm{r}}\mid=\sqrt{\mathrm{5}.\mathrm{4}\:}\:.\mathrm{If}\:\mathrm{X}=\:\mathrm{7}\boldsymbol{\mathrm{p}}+\mathrm{5}\boldsymbol{\mathrm{q}}+\mathrm{7}\boldsymbol{\mathrm{r}}\:\mathrm{and} \\ $$$$\mathrm{Y}=\mathrm{2}\boldsymbol{\mathrm{p}}+\mathrm{3}\boldsymbol{\mathrm{q}}−\mathrm{5}\boldsymbol{\mathrm{r}}\:\mathrm{are}\:\mathrm{perpendicular},\:\mathrm{find}\mid\boldsymbol{\mathrm{p}}\mid. \\ $$

Question Number 111002    Answers: 1   Comments: 1

(√(bemath)) ⇒ sin 14°+cos 14°tan 38°−1=?

$$\sqrt{\mathrm{bemath}} \\ $$$$\Rightarrow\:\mathrm{sin}\:\mathrm{14}°+\mathrm{cos}\:\mathrm{14}°\mathrm{tan}\:\mathrm{38}°−\mathrm{1}=? \\ $$

Question Number 111001    Answers: 0   Comments: 1

Two numbers a and b are chosen at random from the set of first 30 natural numbers. The probability that a^2 −b^2 is divisible by 3 is

$$\mathrm{Two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{chosen}\:\mathrm{at} \\ $$$$\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{first}\:\mathrm{30}\:\mathrm{natural} \\ $$$$\mathrm{numbers}.\:\mathrm{The}\:\mathrm{probability}\:\mathrm{that}\:{a}^{\mathrm{2}} −{b}^{\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}\:\mathrm{is} \\ $$

Question Number 110993    Answers: 1   Comments: 1

Question Number 111092    Answers: 2   Comments: 4

(√(bemath)) lim_(x→0) ((arctan x)/(arc sin x−x))

$$\:\:\sqrt{\mathrm{bemath}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{arctan}\:\mathrm{x}}{\mathrm{arc}\:\mathrm{sin}\:\mathrm{x}−\mathrm{x}} \\ $$

Question Number 110988    Answers: 3   Comments: 0

Evaluate without using L′hopital′s rule lim_(x→4) (((√x)−2)/(x−4))

$$\:\mathrm{Evaluate}\:\mathrm{without}\:\mathrm{using}\:\mathrm{L}'\mathrm{hopital}'\mathrm{s}\:\mathrm{rule} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\:\frac{\sqrt{{x}}−\mathrm{2}}{{x}−\mathrm{4}} \\ $$

Question Number 110984    Answers: 1   Comments: 3

GCD of two unequal numbers can′t exceed their absolute difference. Prove.

$$\mathrm{GCD}\:{of}\:{two}\:{unequal}\:\:{numbers}\:{can}'{t}\: \\ $$$${exceed}\:{their}\:{absolute} \\ $$$${difference}.\:\:{Prove}. \\ $$

Question Number 110980    Answers: 2   Comments: 2

Question Number 110964    Answers: 1   Comments: 0

solve ∫_0 ^1 ((x^2 lnx)/((1+x^2 )^3 ))dx

$${solve}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} \mathrm{ln}{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }{dx} \\ $$

Question Number 111017    Answers: 2   Comments: 0

(√(bemath)) ∫ (dx/( ((4−((3−2x))^(1/(3 )) ))^(1/(4 )) )) ?

$$\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\int\:\frac{\mathrm{dx}}{\:\sqrt[{\mathrm{4}\:}]{\mathrm{4}−\sqrt[{\mathrm{3}\:}]{\mathrm{3}−\mathrm{2x}}}}\:? \\ $$

Question Number 110954    Answers: 1   Comments: 0

verify the formulae Σ_(n=−∞) ^(+∞) (1/((na +1)^p )) =−(π/a^n ) lim_(z→−(1/a)) (1/((p−1)!)){cotan(πz)}^((p−1)) inthis case 1) a =1 and p=2 2) a=2 and p=2 3)a=2 and p=3 4) a=3 and p=2

$$\mathrm{verify}\:\mathrm{the}\:\mathrm{formulae} \\ $$$$\sum_{\mathrm{n}=−\infty} ^{+\infty} \:\frac{\mathrm{1}}{\left(\mathrm{na}\:+\mathrm{1}\right)^{\mathrm{p}} }\:=−\frac{\pi}{\mathrm{a}^{\mathrm{n}} }\:\mathrm{lim}_{\mathrm{z}\rightarrow−\frac{\mathrm{1}}{\mathrm{a}}} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{p}−\mathrm{1}\right)!}\left\{\mathrm{cotan}\left(\pi\mathrm{z}\right)\right\}^{\left(\mathrm{p}−\mathrm{1}\right)} \\ $$$$\left.\mathrm{inthis}\:\mathrm{case}\:\:\mathrm{1}\right)\:\:\mathrm{a}\:=\mathrm{1}\:\mathrm{and}\:\mathrm{p}=\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}=\mathrm{2}\:\:\:\mathrm{and}\:\mathrm{p}=\mathrm{2} \\ $$$$\left.\mathrm{3}\right)\mathrm{a}=\mathrm{2}\:\mathrm{and}\:\mathrm{p}=\mathrm{3} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{a}=\mathrm{3}\:\mathrm{and}\:\mathrm{p}=\mathrm{2} \\ $$

Question Number 110953    Answers: 0   Comments: 1

Question Number 110951    Answers: 2   Comments: 0

Question Number 110948    Answers: 1   Comments: 6

(√(bemath)) If each point on the line 3x+4y=2 is transformed by matrix M= (((2 0)),((0 1)) ) , the image is a line ___

$$\:\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\mathrm{If}\:\mathrm{each}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:\mathrm{3x}+\mathrm{4y}=\mathrm{2} \\ $$$$\mathrm{is}\:\mathrm{transformed}\:\mathrm{by}\:\mathrm{matrix}\:\mathrm{M}=\begin{pmatrix}{\mathrm{2}\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\mathrm{1}}\end{pmatrix}\:,\:\mathrm{the} \\ $$$$\mathrm{image}\:\mathrm{is}\:\mathrm{a}\:\mathrm{line}\:\_\_\_ \\ $$

Question Number 110944    Answers: 5   Comments: 3

■(√(bemath))★ (1)If (√a) −(√b) = 20 , a,b∈R , find maximum value of a−5b ? (2)lim_(x→4) (((√x)−(√(3(√x)−2)))/(x^2 −16)) ? (3)∫ ((tan (ln x) tan (ln ((x/2))))/x) dx (4)((((√(3x−7)))^2 −2)/(x−3)) ≤ ((3−((√x))^2 )/(x−3))

$$\:\:\:\blacksquare\sqrt{\mathrm{bemath}}\bigstar \\ $$$$\left(\mathrm{1}\right)\mathrm{If}\:\sqrt{\mathrm{a}}\:−\sqrt{\mathrm{b}}\:=\:\mathrm{20}\:,\:\mathrm{a},\mathrm{b}\in\mathbb{R}\:,\:\mathrm{find}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{a}−\mathrm{5b}\:? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{3}\sqrt{\mathrm{x}}−\mathrm{2}}}{\mathrm{x}^{\mathrm{2}} −\mathrm{16}}\:? \\ $$$$\left(\mathrm{3}\right)\int\:\frac{\mathrm{tan}\:\left(\mathrm{ln}\:\mathrm{x}\right)\:\mathrm{tan}\:\left(\mathrm{ln}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{x}}\:\mathrm{dx} \\ $$$$\left(\mathrm{4}\right)\frac{\left(\sqrt{\mathrm{3x}−\mathrm{7}}\right)^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}−\mathrm{3}}\:\leqslant\:\frac{\mathrm{3}−\left(\sqrt{\mathrm{x}}\right)^{\mathrm{2}} }{\mathrm{x}−\mathrm{3}} \\ $$

Question Number 110939    Answers: 2   Comments: 1

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