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Question Number 139612 Answers: 1 Comments: 0

show that ∫_0 ^( ∞) ((cos((√x)))/(e^(2π(√x)) −1))dx = 1−(e/((e−1)^2 ))

$s h o w t h a t$ $\int_{0}^{\infty} \frac{c o s (\sqrt{x})}{e^{2 π \sqrt{x}} - 1} d x = 1 - \frac{e}{{(e - 1)}^{2}}$

Question Number 139609 Answers: 2 Comments: 0

If tan 14° = x then tan 18° =?

$If \tan 14 ° = x then \tan 18 ° = ?$

Question Number 139608 Answers: 2 Comments: 0

find the first root (−8i)^(1/2)

$f i n d t h e f i r s t r o o t {(- 8 i)}^{\frac{1}{2}}$

Question Number 139601 Answers: 0 Comments: 1

Question Number 139600 Answers: 1 Comments: 0

Question Number 139599 Answers: 1 Comments: 0

Question Number 139598 Answers: 0 Comments: 0

L=lim_(x→0) ((6sin(sin(sin(sinx)))−6x+5x^3 )/x^5 )

$L = \lim_{x \to 0} \frac{6 \sin (\sin (\sin (sinx))) - 6 x + {5 x}^{3}}{x^{5}}$

Question Number 139592 Answers: 2 Comments: 0

Given f(x)=(√(2+x^2 −x)) +(√(2−x^2 )) If (g○f)(x) = 2x+1 then g^(−1) (−1)=?

$Given f (x) = \sqrt{2 + x^{2} - x} + \sqrt{2 - x^{2}}$ $If (g \circ f) (x) = 2 x + 1 then g^{- 1} (- 1) = ?$

Question Number 139590 Answers: 1 Comments: 0

If a, b and c are integers not all simultaneously equal and w≠1 is a cube root of unity, then the minimum value of ∣a+bw+cw^2 ∣ is (A) 0 (B) 1 (C) (√3)/2 (D) 1/2

$If a, b and c are integers not all simultaneously equal$ $and w \neq 1 is a cube root of unity, then the minimum$ $value of ∣ a + b w + {c w}^{2} ∣ is$ $(A) 0 (B) 1 (C) \sqrt{3} / 2 (D) 1 / 2$

Question Number 139589 Answers: 0 Comments: 0

Question Number 139583 Answers: 0 Comments: 0

Question Number 139582 Answers: 0 Comments: 0

Bug in version 2.265 1. the cursor keys ↑↓ seem not to work. 2. when inserting new lines with Enter key, the cursor doesn′t move.

$B u g i n v e r s i o n 2.265$ $1 . t h e c u r s o r k e y s ↑↓ s e e m n o t t o w o r k .$ $2 . w h e n i n s e r t i n g n e w l i n e s w i t h$ $E n t e r k e y, t h e c u r s o r {d o e s n}^{'} t m o v e .$

Question Number 139587 Answers: 1 Comments: 0

Question Number 139577 Answers: 1 Comments: 0

Solve for real number(((x−1)^2 )/2)+(((y−2)^2 )/4)+(((z−3)^2 )/6)+3=∣x−1∣+∣y−2∣+∣z−3∣

$Solve for real number \frac{{(x - 1)}^{2}}{2} + \frac{{(y - 2)}^{2}}{4} + \frac{{(z - 3)}^{2}}{6} + 3 =∣ x - 1 ∣ + ∣ y - 2 ∣ + ∣ z - 3 ∣$

Question Number 139561 Answers: 1 Comments: 0

Near the shore a fisherman jumps out of his boat with a velocity of 5.00 ms^(−1) and lands on the shore 0.650 afterwards.The boat moves backwards. The respective masses of the fisherman and the boat are 85.0 kg and 165 kg and the frictional force between the boat and water is negligible. What will be the distance between the fisherman and the both at the instant when he just lands on the shore?

$Near the shore a fisherman jumps out of his boat with a velocity$ $of 5.00 {ms}^{- 1} and lands on the shore 0.650 afterwards . The boat$ $moves backwards . The respective masses of the fisherman and$ $the boat are 85.0 kg and 165 kg and the frictional force between$ $the boat and water is negligible . What will be the distance between$ $the fisherman and the both at the instant when he just lands on the$ $shore ?$

Question Number 139560 Answers: 0 Comments: 0

Σ_(n_1 +2n_2 +3n_3 +..+rn_r =n) ^n (1/(n_1 !n_2 !n_3 !..n_r !1^n_1 2^n_2 3^n_3 4^n_4 ...r^n_r ))=1 0≥n_1 ,n_2 ,n_3 ,..≥n Prove the above identity

$\underset{n_{1} + 2 n_{2} + 3 n_{3} + . . + {r n}_{r} = n}{\sum^{n}} \frac{1}{n_{1}! n_{2}! n_{3}! . . n_{r}! 1^{n_{1}} 2^{n_{2}} 3^{n_{3}} 4^{n_{4}} . . . r^{n_{r}}} = 1$ $0 ⩾ n_{1}, n_{2}, n_{3}, . . ⩾ n$ $P r o v e t h e a b o v e i d e n t i t y$

Question Number 139556 Answers: 0 Comments: 0

........ advanced ... ... ... calculus........ Φ= lim_(n→∞) {∫_1 ^( n) (x/([x]^2 )) dx −ψ(n+1)}=? solution: Φ_n =∫_1 ^( n) (x/([x]^2 )) dx=Σ_(k=1) ^(n−1) ∫_k ^( k+1) (x/k^2 ) dx = (1/2)Σ_(k=1) ^(n−1) (1/k^2 )(2k+1)=Σ_(k=1) ^(n−1) (1/k)+(1/2)Σ_(k=1) ^(n−1) (1/k^2 ) Φ = lim_(n→∞) (Φ_n −ψ(n+1)) = (π^2 /(12)) +lim_(n→∞) (Σ_(k=1) ^(n−1) (1/k)−ψ(n+1)) =_(2 : ψ(n+1)= H_n −γ ) ^(1 :ψ (n+1) := (1/n) +ψ(n)) (π^2 /(12))+lim_(n→∞) (Σ_(k=1) ^(n−1) (1/k)−H_n +γ) ∴ Φ := (π^2 /(12)) +γ −lim_(n→∞) ((1/n)) ......... Φ:=(1/2) ζ (2) +γ γ :: Euler− Mascheroni constant...

$. . . . . . . . a d v a n c e d . . . . . . . . . c a l c u l u s . . . . . . . .$ $Φ = {l i m}_{n \to \infty} {\int_{1}^{n} \frac{x}{{[x]}^{2}} d x - ψ (n + 1)} = ?$ $s o l u t i o n :$ $Φ_{n} = \int_{1}^{n} \frac{x}{{[x]}^{2}} d x = \underset{k = 1}{\sum^{n - 1}} \int_{k}^{k + 1} \frac{x}{k^{2}} d x$ $= \frac{1}{2} \underset{k = 1}{\sum^{n - 1}} \frac{1}{k^{2}} (2 k + 1) = \underset{k = 1}{\sum^{n - 1}} \frac{1}{k} + \frac{1}{2} \underset{k = 1}{\sum^{n - 1}} \frac{1}{k^{2}}$ $Φ = {l i m}_{n \to \infty} (Φ_{n} - ψ (n + 1))$ $= \frac{π^{2}}{12} + {l i m}_{n \to \infty} (\underset{k = 1}{\sum^{n - 1}} \frac{1}{k} - ψ (n + 1))$ $\underset{2 : ψ (n + 1) = H_{n} - γ}{\overset{1 : ψ (n + 1) := \frac{1}{n} + ψ (n)}{=}} \frac{π^{2}}{12} + {l i m}_{n \to \infty} (\underset{k = 1}{\sum^{n - 1}} \frac{1}{k} - H_{n} + γ)$ $∴ Φ := \frac{π^{2}}{12} + γ - {l i m}_{n \to \infty} (\frac{1}{n})$ $. . . . . . . . . Φ := \frac{1}{2} ζ (2) + γ$ $γ :: E u l e r - M a s c h e r o n i c o n s t a n t . . .$

Question Number 139555 Answers: 1 Comments: 0

Π_(i=0) ^5 (1−cotan(20+i)) =^? 8

$\underset{i = 0}{\prod^{5}} (1 - c o t a n (20 + i)) \overset{?}{=} 8$

Question Number 139548 Answers: 1 Comments: 0

Question Number 139545 Answers: 1 Comments: 0

Question Number 139544 Answers: 2 Comments: 0

Question Number 139532 Answers: 1 Comments: 4

(1/3) − ((x−3)/9) + (((x−3)^2 )/(27)) + ... =?

$\frac{1}{3} - \frac{x - 3}{9} + \frac{{(x - 3)}^{2}}{27} + . . . = ?$

Question Number 139530 Answers: 1 Comments: 0

.......advanced calculus...... prove that:: lim_(n→∞) {(((−1)^(n+1) n^(n+1) )/(n!)) (d^( n) /dx^n )(((ln(x))/x))∣_(x=n) }=γ γ : euler −mascheroni constant

$. . . . . . . a d v a n c e d c a l c u l u s . . . . . .$ $p r o v e t h a t ::$ ${l i m}_{n \to \infty} {\frac{{(- 1)}^{n + 1} n^{n + 1}}{n!} \frac{d^{n}}{{d x}^{n}} (\frac{l n (x)}{x}) ∣_{x = n}} = γ$ $γ : e u l e r - m a s c h e r o n i c o n s t a n t$

Question Number 139525 Answers: 0 Comments: 3

Question Number 139524 Answers: 1 Comments: 0

......advanced calculus..... prove that: i::𝛗=∫_0 ^( ∞) ((1/2)e^(−2x) −(1/(1+e^x )))(1/x) dx=log((1/( (√π))) ) ii::∫_0 ^( ∞) ((1/2)−(1/(1+e^(−x) )))(e^(−2x) /x)dx=log(((√π)/2))

$. . . . . . a d v a n c e d c a l c u l u s . . . . .$ $p r o v e t h a t :$ $i :: ϕ = \int_{0}^{\infty} (\frac{1}{2} e^{- 2 x} - \frac{1}{1 + e^{x}}) \frac{1}{x} d x = l o g (\frac{1}{\sqrt{π}})$ $i i :: \int_{0}^{\infty} (\frac{1}{2} - \frac{1}{1 + e^{- x}}) \frac{e^{- 2 x}}{x} d x = l o g (\frac{\sqrt{π}}{2})$

Question Number 141965 Answers: 1 Comments: 1

lim_(x→0) ((x^n sin^n x)/(x^n −sin^n x)) is non-zero finite, then find n?

$\lim_{x \to 0} \frac{x^{n} \sin^{n} x}{x^{n} - \sin^{n} x} i s n o n - z e r o f i n i t e,$ $t h e n f i n d n ?$

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