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Question Number 111043 Answers: 0 Comments: 0

solve the integral ∫_0 ^∞ (dx/((√x)(1+x^2 +x^4 )^(3/4) ))dx

$${solve}\:{the}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\sqrt{{x}}\left(\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} \right)^{\frac{\mathrm{3}}{\mathrm{4}}} }{dx} \\ $$

Question Number 111039 Answers: 1 Comments: 0

solve ∫_0 ^1 (√(1+x^6 ))dx mr abbo your question

$${solve}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+{x}^{\mathrm{6}} }{dx} \\ $$$${mr}\:\:{abbo}\:{your}\:{question}\: \\ $$

Question Number 111035 Answers: 1 Comments: 0

If b∈Z^+ ∀ both the roots of equation x^2 −bx+132=0 are integers. Find (1)the largest possible value of b (2)the smallest possible value of b.

$$\mathrm{If}\:{b}\in\mathbb{Z}^{+} \:\forall\:\mathrm{both}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$${x}^{\mathrm{2}} −{bx}+\mathrm{132}=\mathrm{0}\:\mathrm{are}\:\mathrm{integers}. \\ $$$$\mathrm{Find} \\ $$$$\left(\mathrm{1}\right)\mathrm{the}\:\mathrm{largest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{b} \\ $$$$\left(\mathrm{2}\right)\mathrm{the}\:\mathrm{smallest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:{b}. \\ $$

Question Number 111029 Answers: 1 Comments: 0

Question Number 111083 Answers: 2 Comments: 0

(√(bemath)) (1)∫ (dx/(3sin x+sin^3 x)) (2) lim_(x→∞) x(5^(1/x) −1) (3) find the asymptotes (x^2 /a^2 ) − (y^2 /b^2 ) = 1

$$\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{dx}}{\mathrm{3sin}\:\mathrm{x}+\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}} \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{x}\left(\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{x}}} \:−\mathrm{1}\right)\: \\ $$$$\left(\mathrm{3}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{asymptotes}\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} }\:−\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{b}^{\mathrm{2}} }\:=\:\mathrm{1}\: \\ $$

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

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

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