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Question Number 35051    Answers: 0   Comments: 1

calculate Σ_(n=0) ^∞ ((n+3)/(2n+1))x^n

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{n}+\mathrm{3}}{\mathrm{2}{n}+\mathrm{1}}{x}^{{n}} \\ $$$$ \\ $$

Question Number 35050    Answers: 0   Comments: 0

calculate Σ_(n=0) ^∞ (((−1)^n x^(2n+1) )/(4n^2 −1))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} \:\:{x}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}} \\ $$

Question Number 35049    Answers: 1   Comments: 1

let A_n = ∫_(1/n) ^n (1+(1/x^2 ))arctanx dx 1) calculate A_n 2) find lim_(n→+∞) A_n

$${let}\:{A}_{{n}} \:=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 35048    Answers: 0   Comments: 0

find ∫ (dx/(cos(sinx)))

$${find}\:\int\:\:\:\:\:\frac{{dx}}{{cos}\left({sinx}\right)} \\ $$

Question Number 35046    Answers: 0   Comments: 0

find F(x)= ∫_0 ^π ln( 1+x sin^2 t)dt with ∣x∣<1 2) calculate ∫_0 ^π ln(1+(1/2)sin^2 t)dt

$${find}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\:\mathrm{1}+{x}\:{sin}^{\mathrm{2}} {t}\right){dt}\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{sin}^{\mathrm{2}} {t}\right){dt} \\ $$

Question Number 35045    Answers: 0   Comments: 0

find f(x)=∫_0 ^∞ ((arctan(xt))/(1+t^2 ))dt .

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{arctan}\left({xt}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:\:. \\ $$

Question Number 35044    Answers: 1   Comments: 1

1)find ∫ (√(1+t^2 )) dt 2) calculate ∫_1 ^(√3) (√(1+t^2 )) dt

$$\left.\mathrm{1}\right){find}\:\int\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:\:{dt} \\ $$

Question Number 35043    Answers: 1   Comments: 0

let t>0 and F(t) =∫_0 ^∞ ((sin(x^2 ) e^(−tx^2 ) )/x^2 )dx calculate (dF/dt)(t).

$${let}\:{t}>\mathrm{0}\:{and}\:{F}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)\:{e}^{−{tx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }{dx} \\ $$$${calculate}\:\frac{{dF}}{{dt}}\left({t}\right). \\ $$

Question Number 35041    Answers: 0   Comments: 1

find the value of Σ_(n=1) ^∞ ((n+1)/(n^3 (n+1)^2 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}+\mathrm{1}}{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 35040    Answers: 0   Comments: 0

find the value of Σ_(n=1) ^∞ (x^n /(n(n+1)(2n−1))) 2) find the value of Σ_(n=1) ^∞ (((−1)^n )/(2^n n(n+1)(2n−1)))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\:\frac{{x}^{{n}} }{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{{n}} {n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)} \\ $$

Question Number 35023    Answers: 1   Comments: 1

Question Number 35021    Answers: 0   Comments: 1

Find the points at which the given function is discontinuous : f(x)= ∣x∣ sgn (x^3 −x) .

$${Find}\:{the}\:{points}\:{at}\:{which}\:{the}\:{given} \\ $$$${function}\:{is}\:{discontinuous}\:: \\ $$$${f}\left({x}\right)=\:\mid{x}\mid\:{sgn}\:\left({x}^{\mathrm{3}} −{x}\right)\:. \\ $$

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 ))))))))}

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

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