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AllQuestion and Answers: Page 1736

Question Number 35018 Answers: 1 Comments: 0

∫∫∫((dxdydz)/((x+y+z+1)^3 )) bounded by the coordinate planes and the plane x+y+z=1 .

$$\int\int\int\frac{{dxdydz}}{\left({x}+{y}+{z}+\mathrm{1}\right)^{\mathrm{3}} }\:\:\:{bounded}\:{by}\:{the} \\ $$$${coordinate}\:{planes}\:{and}\:{the}\:{plane} \\ $$$${x}+{y}+{z}=\mathrm{1}\:. \\ $$

Question Number 35015 Answers: 2 Comments: 0

∫(x^2 /((1+x^3 )^2 ))dx

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

Question Number 35005 Answers: 2 Comments: 0

Find remainder when 1!+2!+3!+...+99!+100! is divided by 12.

$$\mathrm{Find}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\:\:\:\:\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+...+\mathrm{99}!+\mathrm{100}! \\ $$$$\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{12}. \\ $$

Question Number 35004 Answers: 3 Comments: 0

Prove that 41 divides 2^(20) −1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{41}\:\mathrm{divides}\:\mathrm{2}^{\mathrm{20}} −\mathrm{1} \\ $$

Question Number 34992 Answers: 1 Comments: 1

∫_0 ^π ((cos(x))/(1+2sin(2x)))dx

$$\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{\mathrm{cos}\left({x}\right)}{\mathrm{1}+\mathrm{2sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 34988 Answers: 1 Comments: 1

Question Number 34982 Answers: 2 Comments: 2

lim_(x→1) ((nx^(n+1) −(n+1)x^n +1)/((e^x −e)sin πx)) = ?

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{{nx}^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right){x}^{{n}} +\mathrm{1}}{\left({e}^{{x}} −{e}\right)\mathrm{sin}\:\pi{x}}\:=\:? \\ $$

Question Number 34980 Answers: 2 Comments: 0

If cos45° =(1/(√2)) . Find cos45.1°

$${If}\:{cos}\mathrm{45}°\:=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:.\:{Find}\:{cos}\mathrm{45}.\mathrm{1}° \\ $$

Question Number 34956 Answers: 3 Comments: 2

Question Number 34954 Answers: 1 Comments: 1

Question Number 34952 Answers: 0 Comments: 2

Question Number 34951 Answers: 1 Comments: 0

lim_(n→∞) (((e×a)/n))^n = ? Here aε R^+

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{e}×{a}}{{n}}\right)^{{n}} \:=\:?\: \\ $$$${Here}\:{a}\epsilon\:\mathbb{R}^{+} \\ $$

Question Number 34930 Answers: 1 Comments: 3

A committee of 4 is to be formed from 4 principals and 5 students.In how many ways can this be done if a particular student and a principal must be in the committee.

$${A}\:{committee}\:{of}\:\mathrm{4}\:{is}\:{to}\:{be}\:{formed} \\ $$$${from}\:\mathrm{4}\:{principals} \\ $$$${and}\:\mathrm{5}\:{students}.{In}\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}\:{if}\:{a}\:{particular} \\ $$$${student}\:{and}\:{a}\:{principal}\:{must}\:{be} \\ $$$${in}\:{the}\:{committee}. \\ $$$$ \\ $$

Question Number 34921 Answers: 1 Comments: 0

lim_(x→∞) { (x/(x+(((x)^(1/3) )/(x+ (((x)^(1/3) )/(x+(((x)^(1/3) )/(......... infinity ))))))))}

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left\{\:\frac{{x}}{{x}+\frac{\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }{{x}+\:\frac{\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }{{x}+\frac{\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} }{.........\:{infinity}\:}}}}\right\} \\ $$

Question Number 34913 Answers: 0 Comments: 6

let f(x)= (3/(2+cosx)) developp f ar fourier serie.

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{3}}{\mathrm{2}+{cosx}}\:\:{developp}\:{f}\:{ar}\:{fourier}\:{serie}. \\ $$

Question Number 34912 Answers: 1 Comments: 0

let f(x,y,z) =(x^2 +y^2 +z^2 )^α with α∈R 1) calculate Δf 2) find α in order to have Δf=0

$${let}\:{f}\left({x},{y},{z}\right)\:=\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)^{\alpha} \:\:\:\:\:{with}\:\alpha\in{R} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\Delta{f} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\alpha\:{in}\:{order}\:{to}\:{have}\:\Delta{f}=\mathrm{0} \\ $$

Question Number 34911 Answers: 1 Comments: 1

find ∫_2 ^3 ((2x^2 +3)/((x−1)^2 (x^2 +1))) dx

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

Question Number 34910 Answers: 0 Comments: 1

find J_(n,p) =∫_0 ^∞ x^n e^(−(x^2 /p)) dx with p>0 and n integr

$${find}\:{J}_{{n},{p}} \:=\int_{\mathrm{0}} ^{\infty} \:\:{x}^{{n}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{{p}}} \:\:{dx}\:\:{with}\:{p}>\mathrm{0}\:{and}\:{n}\:{integr} \\ $$

Question Number 35037 Answers: 0 Comments: 0

prove that ∀ n∈N Σ_(k=0) ^(2n) (−1)^k ( C_(2n) ^k )^2 =(−1)^n C_(2n) ^n .

$${prove}\:{that}\:\forall\:{n}\in{N} \\ $$$$\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\left(−\mathrm{1}\right)^{{k}} \left(\:{C}_{\mathrm{2}{n}} ^{{k}} \right)^{\mathrm{2}} \:=\left(−\mathrm{1}\right)^{{n}} \:{C}_{\mathrm{2}{n}} ^{{n}} \:\:. \\ $$

Question Number 35036 Answers: 1 Comments: 0

cslculate Σ_(k=1) ^n k^2 (n+1−k)

$${cslculate}\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}^{\mathrm{2}} \left({n}+\mathrm{1}−{k}\right) \\ $$

Question Number 35035 Answers: 1 Comments: 0

calculate u_n = Σ_(k=1) ^n (k/((k+1)!))

$${calculate}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}}{\left({k}+\mathrm{1}\right)!} \\ $$

Question Number 35032 Answers: 0 Comments: 0

Question Number 34892 Answers: 0 Comments: 0

Question Number 34891 Answers: 1 Comments: 0

Question Number 34888 Answers: 1 Comments: 1

Question Number 34901 Answers: 0 Comments: 3

∫_(−π/2) ^(+π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =[−((2cos^((2n+1)/2) x)/(2n+1))]_(−π/2) ^(+π/2) =0? What is the mistake in above? ∫_(−π/2) ^(+π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =2∫_0 ^(π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =(4/(2n+1)) (this is correct answer)

$$\int_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\left[−\frac{\mathrm{2cos}^{\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}}} {x}}{\mathrm{2}{n}+\mathrm{1}}\right]_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} =\mathrm{0}? \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mistake}\:\mathrm{in}\:\mathrm{above}? \\ $$$$\int_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\frac{\mathrm{4}}{\mathrm{2}{n}+\mathrm{1}}\:\left(\mathrm{this}\:\mathrm{is}\:\mathrm{correct}\:\mathrm{answer}\right) \\ $$

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