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Question Number 168800 Answers: 0 Comments: 1

If the function f is continuous in [a,b] express lim_(n→∞) (1/n)Σ_(k=1) ^n f((k/n)) as a definite integral.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{function}\:{f}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{in}\:\left[{a},{b}\right] \\ $$$$\mathrm{express}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{f}\left(\frac{{k}}{{n}}\right)\:\mathrm{as}\:\mathrm{a}\:\mathrm{definite} \\ $$$$\mathrm{integral}. \\ $$

Question Number 168799 Answers: 0 Comments: 0

If the function f is continuous in [a,b] prove that lim_(x→∞ ) ((b−a)/n)Σ_(k=1) ^n f(a+((k(b−a))/n))=∫_a ^b f(x)dx

$$\mathrm{If}\:\mathrm{the}\:\mathrm{function}\:{f}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{in} \\ $$$$\left[{a},{b}\right]\: \\ $$$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\:\underset{{x}\rightarrow\infty\:} {\mathrm{lim}}\frac{{b}−{a}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{f}\left({a}+\frac{{k}\left({b}−{a}\right)}{{n}}\right)=\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$

Question Number 168788 Answers: 0 Comments: 1

Question Number 168787 Answers: 0 Comments: 1

Question Number 168781 Answers: 0 Comments: 1

Question Number 168776 Answers: 1 Comments: 4

Question Number 168823 Answers: 2 Comments: 0

Q#168480 reposted. n^2 +n+109=x^2 x∈Z, n(∈Z^+ )=?

$$\boldsymbol{\mathrm{Q}}#\mathrm{168480}\:\mathrm{reposted}. \\ $$$$\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}+\mathrm{109}=\boldsymbol{\mathrm{x}}^{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{x}}\in\mathbb{Z},\:\boldsymbol{\mathrm{n}}\left(\in\mathbb{Z}^{+} \right)=? \\ $$

Question Number 168772 Answers: 1 Comments: 1

Question Number 168771 Answers: 0 Comments: 0

Question Number 168762 Answers: 0 Comments: 2

Question Number 168755 Answers: 0 Comments: 5

∫(1/(x+(√(1−x)))) dx

$$\int\frac{\mathrm{1}}{{x}+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$

Question Number 168753 Answers: 0 Comments: 1

∫(dx/( (√(sin^(−1) (x)))))=?

$$\int\frac{{dx}}{\:\sqrt{{sin}^{−\mathrm{1}} \left({x}\right)}}=? \\ $$

Question Number 168745 Answers: 1 Comments: 1

∫(1/(x+(√(x−1)))) dx = ??

$$\int\frac{\mathrm{1}}{{x}+\sqrt{{x}−\mathrm{1}}}\:{dx}\:=\:?? \\ $$

Question Number 168744 Answers: 3 Comments: 0

Resolve 1) ∫((√(1+cos x))/(sin x))dx 2) ∫(dx/(1+((x+1))^(1/3) )) 3) ∫((xtan x)/(cos^4 x))dx 4) ∫(dx/(1+(√x)+(√(1+x))))

$${Resolve} \\ $$$$\left.\mathrm{1}\right)\:\int\frac{\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}{\mathrm{sin}\:{x}}{dx} \\ $$$$\left.\mathrm{2}\right)\:\int\frac{{dx}}{\mathrm{1}+\sqrt[{\mathrm{3}}]{{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{3}\right)\:\int\frac{{x}\mathrm{tan}\:{x}}{\mathrm{cos}\:^{\mathrm{4}} {x}}{dx} \\ $$$$\left.\mathrm{4}\right)\:\int\frac{{dx}}{\mathrm{1}+\sqrt{{x}}+\sqrt{\mathrm{1}+{x}}} \\ $$

Question Number 168743 Answers: 1 Comments: 1

∫_0 ^∞ ((√t)/(1+t^2 ))dt FAILED TO CALCULATE

$$\int_{\mathrm{0}} ^{\infty} \frac{\sqrt{{t}}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\mathrm{FAILED}\:\mathrm{TO}\:\mathrm{CALCULATE} \\ $$$$ \\ $$

Question Number 168742 Answers: 1 Comments: 0

Resolve y=xy′+a(√(1+(y′)^2 ))

$${Resolve} \\ $$$${y}={xy}'+{a}\sqrt{\mathrm{1}+\left(\mathrm{y}'\right)^{\mathrm{2}} } \\ $$

Question Number 168737 Answers: 1 Comments: 1

Question Number 168734 Answers: 0 Comments: 0

Question Number 168732 Answers: 2 Comments: 0

Question Number 168723 Answers: 0 Comments: 2

Resolve (x+5)^5 y^(′′) =1

$${Resolve}\: \\ $$$$\left({x}+\mathrm{5}\right)^{\mathrm{5}} {y}^{''} =\mathrm{1} \\ $$

Question Number 168722 Answers: 0 Comments: 1

Resolve x^2 y^(′′) +xy^′ +y=1

$${Resolve}\: \\ $$$${x}^{\mathrm{2}} {y}^{''} +{xy}^{'} +{y}=\mathrm{1} \\ $$

Question Number 168721 Answers: 1 Comments: 0

Question Number 168710 Answers: 0 Comments: 4

Question Number 168709 Answers: 0 Comments: 3

Question Number 168707 Answers: 1 Comments: 1

Question Number 168698 Answers: 3 Comments: 0

3f(x)+2f(((x+59)/(x−1)))=10x+30 for all rael x=1 faind volue of f(7)=?

$$\mathrm{3}{f}\left({x}\right)+\mathrm{2}{f}\left(\frac{{x}+\mathrm{59}}{{x}−\mathrm{1}}\right)=\mathrm{10}{x}+\mathrm{30}\:{for}\:{all} \\ $$$${rael}\:\:{x}\cancel{=}\mathrm{1}\:{faind}\:{volue}\:{of} \\ $$$${f}\left(\mathrm{7}\right)=? \\ $$$$ \\ $$

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