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Question Number 115627 Answers: 0 Comments: 1

Question Number 115621 Answers: 2 Comments: 0

... advanced calculus... evaluate :: show that lim_(n→∞) (1/n)[cos^(2p) (π/(2n))+cos^(2p) ((2π)/(2n))+cos^(2p) ((3π)/(2n))......cos^(2p) (π/2)] =Π_(r=1) ^p ((p+r)/(4r))

$$ \\ $$$$\:\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}...\: \\ $$$$\:\:\:\:\:\:\:{evaluate}\::: \\ $$$${show}\:{that}\:{lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\left[{cos}^{\mathrm{2}{p}} \frac{\pi}{\mathrm{2}{n}}+{cos}^{\mathrm{2}{p}} \frac{\mathrm{2}\pi}{\mathrm{2}{n}}+{cos}^{\mathrm{2}{p}} \frac{\mathrm{3}\pi}{\mathrm{2}{n}}......{cos}^{\mathrm{2}{p}} \frac{\pi}{\mathrm{2}}\right]\:=\underset{{r}=\mathrm{1}} {\overset{{p}} {\prod}}\frac{{p}+{r}}{\mathrm{4}{r}} \\ $$

Question Number 115616 Answers: 2 Comments: 0

lim_(x→0) ((((256+tan x))^(1/(8 )) −((1+sin x))^(1/(3 )) −1)/( ((1+tan x))^(1/(6 )) −1)) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{8}\:}]{\mathrm{256}+\mathrm{tan}\:\:{x}}\:−\sqrt[{\mathrm{3}\:}]{\mathrm{1}+\mathrm{sin}\:{x}}\:−\mathrm{1}}{\:\sqrt[{\mathrm{6}\:}]{\mathrm{1}+\mathrm{tan}\:{x}}\:−\mathrm{1}}\:? \\ $$

Question Number 115604 Answers: 6 Comments: 1

(1)lim_(x→(π/2)) ((cos 2x)/(tan x)) =? (2) lim_(x→π) (((√(2 +cos x)) −1)/((π−x)^2 )) = ?

$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{tan}\:{x}}\:=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}\:+\mathrm{cos}\:{x}}\:−\mathrm{1}}{\left(\pi−{x}\right)^{\mathrm{2}} }\:=\:? \\ $$

Question Number 115597 Answers: 4 Comments: 0

Question Number 115595 Answers: 1 Comments: 0

Prove that, for all primes p>3, 13∣10^(2p) −10^p +1

$$\mathrm{Prove}\:\mathrm{that},\:\mathrm{for}\:\mathrm{all}\:\mathrm{primes}\:\mathrm{p}>\mathrm{3}, \\ $$$$\mathrm{13}\mid\mathrm{10}^{\mathrm{2p}} −\mathrm{10}^{\mathrm{p}} +\mathrm{1} \\ $$

Question Number 115594 Answers: 1 Comments: 0

old question, I couldn′t find it: ∫(√(x−(√x)))dx=?

$$\mathrm{old}\:\mathrm{question},\:\mathrm{I}\:\mathrm{couldn}'\mathrm{t}\:\mathrm{find}\:\mathrm{it}: \\ $$$$\int\sqrt{{x}−\sqrt{{x}}}{dx}=? \\ $$

Question Number 115592 Answers: 0 Comments: 1

Question Number 115625 Answers: 0 Comments: 1

Question Number 115575 Answers: 3 Comments: 2

lim_(n→∞) Π_(k=1) ^n (1−(1/(k+1)))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{k}+\mathrm{1}}\right)=? \\ $$

Question Number 115564 Answers: 1 Comments: 0

Question Number 115558 Answers: 2 Comments: 2

... advanced calculus... evaluate :: ∫_0 ^( ∞) ln(1+ax^2 )ln(1+(b/x^2 ))dx m.n.july

$$\:\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}...\: \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\mathrm{1}+{ax}^{\mathrm{2}} \right){ln}\left(\mathrm{1}+\frac{{b}}{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\:\:\:\:\:\:\:{m}.{n}.{july} \\ $$$$ \\ $$

Question Number 115555 Answers: 3 Comments: 1

Given that x,y∈R ∀ x^2 −y^2 =32, (x+y)^4 +(x−y)^4 =4352, Find the value of x^2 +y^2 .

$$\mathrm{Given}\:\mathrm{that}\:{x},{y}\in\mathbb{R}\:\forall\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{32}, \\ $$$$\left({x}+{y}\right)^{\mathrm{4}} +\left({x}−{y}\right)^{\mathrm{4}} =\mathrm{4352},\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} . \\ $$

Question Number 115551 Answers: 2 Comments: 0

lim_(x→2) (((x^4 −4x^3 +5x^2 −4x+4))^(1/(4 )) /( (√(x^2 −3x+2)))) = ?

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{4}\:}]{{x}^{\mathrm{4}} −\mathrm{4}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{4}}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}}\:=\:? \\ $$

Question Number 115544 Answers: 1 Comments: 1

Given x^2 +12(√x) = 5 then x+2(√x) ?

$${Given}\:{x}^{\mathrm{2}} +\mathrm{12}\sqrt{{x}}\:=\:\mathrm{5} \\ $$$${then}\:{x}+\mathrm{2}\sqrt{{x}}\:? \\ $$

Question Number 115541 Answers: 3 Comments: 0

lim_(x→a) (2−(x/a))^(tan (((πx)/(2a)))) =?

$$\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\left(\mathrm{2}−\frac{{x}}{{a}}\right)^{\mathrm{tan}\:\left(\frac{\pi{x}}{\mathrm{2}{a}}\right)} =? \\ $$

Question Number 115539 Answers: 1 Comments: 0

(dy/dx) = ((e^(tan^(−1) (x)) −y)/(1+x^2 ))

$$\frac{{dy}}{{dx}}\:=\:\frac{{e}^{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)} \:−{y}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$

Question Number 115534 Answers: 4 Comments: 1

Let say r^((n)) = Π_(k=0) ^(n−1) (r−k) and r^((0)) =1 With n∈N and r∈R... 1. Show that (n−1−r)^((n)) = (−1)^((n)) (r)^((n)) 2. If m≤n, show that (r^((n)) /r^((m)) )=(r−m)^((n−m)) 3. Espress r^((n+m)) as w^((n)) w′^((m)) 4. Show that (2r)^((2n)) =2^(2n) r^((n)) (r−(1/2))^((n)) Can you help me... please...

$$\mathrm{Let}\:\mathrm{say}\:{r}^{\left({n}\right)} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({r}−{k}\right)\:\mathrm{and}\:{r}^{\left(\mathrm{0}\right)} =\mathrm{1} \\ $$$$\mathrm{With}\:{n}\in\mathbb{N}\:\mathrm{and}\:{r}\in\mathbb{R}... \\ $$$$\mathrm{1}.\:\:\:\mathrm{Show}\:\mathrm{that}\:\left({n}−\mathrm{1}−{r}\right)^{\left({n}\right)} \:=\:\left(−\mathrm{1}\right)^{\left({n}\right)} \left({r}\right)^{\left({n}\right)} \\ $$$$\mathrm{2}.\:\mathrm{If}\:{m}\leqslant{n},\:\mathrm{show}\:\mathrm{that}\:\:\frac{{r}^{\left({n}\right)} }{{r}^{\left({m}\right)} }=\left({r}−{m}\right)^{\left({n}−{m}\right)} \\ $$$$\mathrm{3}.\:\mathrm{Espress}\:{r}^{\left({n}+{m}\right)} \:\mathrm{as}\:{w}^{\left({n}\right)} {w}'^{\left({m}\right)} \\ $$$$\mathrm{4}.\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{2}{r}\right)^{\left(\mathrm{2}{n}\right)} =\mathrm{2}^{\mathrm{2}{n}} {r}^{\left({n}\right)} \left({r}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\left({n}\right)} \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}...\:\mathrm{please}... \\ $$

Question Number 115533 Answers: 0 Comments: 0

Question Number 115531 Answers: 0 Comments: 0

Question Number 115532 Answers: 0 Comments: 0

Question Number 115520 Answers: 1 Comments: 0

Given matrix A = (((a 1 1)),((1 a 1)),((1 1 a)) ) If B = b.A and B is orthogonal determine value of a and b.

$${Given}\:{matrix}\:{A}\:=\:\begin{pmatrix}{{a}\:\:\:\mathrm{1}\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:{a}\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\mathrm{1}\:\:\:\:{a}}\end{pmatrix}\: \\ $$$${If}\:{B}\:=\:{b}.{A}\:{and}\:{B}\:{is}\:{orthogonal}\: \\ $$$${determine}\:{value}\:{of}\:{a}\:{and}\:{b}. \\ $$

Question Number 115516 Answers: 1 Comments: 1

Find the supremum and the infimum of (x/(sin x)) on the interval (0, (π/2) ]

$${Find}\:{the}\:{supremum}\:{and}\:{the}\:{infimum} \\ $$$${of}\:\frac{{x}}{\mathrm{sin}\:{x}}\:{on}\:{the}\:{interval}\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\:\right] \\ $$

Question Number 115508 Answers: 2 Comments: 0

If determinant (((a a^2 1+a^3 )),((b b^2 1+b^3 )),((c c^2 1+c^3 )))= 0 a≠b≠c → { ((a =?)),((b=? )),((c=?)) :}

$${If}\begin{vmatrix}{{a}\:\:\:\:{a}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{1}+{a}^{\mathrm{3}} }\\{{b}\:\:\:\:{b}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{1}+{b}^{\mathrm{3}} }\\{{c}\:\:\:\:{c}^{\mathrm{2}} \:\:\:\:\:\:\:\mathrm{1}+{c}^{\mathrm{3}} }\end{vmatrix}=\:\mathrm{0} \\ $$$${a}\neq{b}\neq{c}\:\rightarrow\begin{cases}{{a}\:=?}\\{{b}=?\:}\\{{c}=?}\end{cases} \\ $$

Question Number 115507 Answers: 2 Comments: 0

.... ...matematical analysis... prove that ::: a>0 :: [((i : ∫_(0 ) ^( ∞) ((sin^2 (ax))/x^(3/2) ) dx= (√(πa)))),((ii: ∫_0 ^( ∞) ((sin^3 (ax))/( (√x))) dx = ((−1+3(√(3 )))/4) (√((π/(6a)) )) )) ] ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:....\:\:\:...{matematical}\:{analysis}...\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}>\mathrm{0}\:::\:\:\:\begin{bmatrix}{{i}\::\:\:\int_{\mathrm{0}\:} ^{\:\infty} \frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:{dx}=\:\sqrt{\pi{a}}}\\{{ii}:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({ax}\right)}{\:\sqrt{{x}}}\:{dx}\:=\:\frac{−\mathrm{1}+\mathrm{3}\sqrt{\mathrm{3}\:}}{\mathrm{4}}\:\sqrt{\frac{\pi}{\mathrm{6}{a}}\:\:}\:}\end{bmatrix} \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 115499 Answers: 0 Comments: 0

form a double integral to represent the area of a plane fiqure bounded by the polar centre

$${form}\:{a}\:{double}\:{integral}\:{to}\:{represent} \\ $$$${the}\:{area}\:{of}\:{a}\:{plane}\:{fiqure}\:{bounded}\:{by} \\ $$$${the}\:{polar}\:{centre} \\ $$

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