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Question Number 40886 Answers: 0 Comments: 0

prove that ∫_0 ^1 ((t^(2p+1) ln(t))/(t^2 −1))dt =(π^2 /(24)) −(1/4)Σ_(k=1) ^p (1/k^2 )

$p r o v e t h a t$ $\int_{0}^{1} \frac{t^{2 p + 1} l n (t)}{t^{2} - 1} d t = \frac{π^{2}}{24} - \frac{1}{4} \sum_{k = 1}^{p} \frac{1}{k^{2}}$

Question Number 40885 Answers: 0 Comments: 1

prove that 1) ∫_0 ^1 ((t^p ln(t))/(t−1))dt =(π^2 /6) −Σ_(k=1) ^p (1/k^2 ) 2) ∫_0 ^1 ((t^(2p) ln(t))/(t^2 −1))dt =(π^2 /8) −Σ_(k=0) ^(p−1) (1/((2k+1)^2 ))

$p r o v e t h a t$ $1) \int_{0}^{1} \frac{t^{p} l n (t)}{t - 1} d t = \frac{π^{2}}{6} - \sum_{k = 1}^{p} \frac{1}{k^{2}}$ $2) \int_{0}^{1} \frac{t^{2 p} l n (t)}{t^{2} - 1} d t = \frac{π^{2}}{8} - \sum_{k = 0}^{p - 1} \frac{1}{{(2 k + 1)}^{2}}$

Question Number 40884 Answers: 2 Comments: 0

1) fond ∫_0 ^1 ((ln(t))/(t^2 −1))dt 2) find ∫_0 ^1 ((ln(t))/(t^4 −1))dt

$1) f o n d \int_{0}^{1} \frac{l n (t)}{t^{2} - 1} d t$ $2) f i n d \int_{0}^{1} \frac{l n (t)}{t^{4} - 1} d t$

Question Number 40883 Answers: 1 Comments: 0

find ∫_0 ^∞ (t^p /(e^t −1))dt with p∈N^★

$f i n d \int_{0}^{\infty} \frac{t^{p}}{e^{t} - 1} d t w i t h p \in N^{★}$

Question Number 40882 Answers: 0 Comments: 0

1)prove that ∀n≥2(n inyegr) x^(2n) −1=(x−1)(x+1)Π_(k=1) ^(n−1) (x^2 −2cos(((kπ)/n))x+1) 2)find the value of ∫_0 ^π ln(x^2 −2xcost +1)dt

$1) p r o v e t h a t \forall n ⩾ 2 (n i n y e g r)$ $x^{2 n} - 1 = (x - 1) (x + 1) \prod_{k = 1}^{n - 1} (x^{2} - 2 c o s (\frac{k π}{n}) x + 1)$ $2) f i n d t h e v a l u e o f \int_{0}^{π} l n (x^{2} - 2 x c o s t + 1) d t$

Question Number 40880 Answers: 0 Comments: 0

prove that Σ_(k=n) ^∞ (1/k^α ) ∼ (1/((α−1)n^(α−1) ))with α>1

$p r o v e t h a t \sum_{k = n}^{\infty} \frac{1}{k^{α}} \sim \frac{1}{(α - 1) n^{α - 1}} w i t h α > 1$

Question Number 40878 Answers: 0 Comments: 0

let u_0 >0 and ∀n∈N u_(n+1) =u_n +(1/u_n ) 1) prove that (u_n )is increasing and lim u_n =+∞ 2)by consideringthe functionϕ(t)=(1/(2t+x)) prove that ∀n∈N Σ_(k=1) ^n (1/(2k+x)) ≤(1/2)ln(1+((2n)/x)) 3)find a equivalent of u_n (n→+∞)

$l e t u_{0} > 0 a n d \forall n \in N$ $u_{n + 1} = u_{n} + \frac{1}{u_{n}}$ $1) p r o v e t h a t (u_{n}) i s i n c r e a s i n g a n d l i m u_{n} = + \infty$ $2) b y c o n s i d e r i n g t h e f u n c t i o n φ (t) = \frac{1}{2 t + x}$ $p r o v e t h a t \forall n \in N \sum_{k = 1}^{n} \frac{1}{2 k + x} ⩽ \frac{1}{2} l n (1 + \frac{2 n}{x})$ $3) f i n d a e q u i v a l e n t o f u_{n} (n \to + \infty)$

Question Number 40876 Answers: 0 Comments: 0

prove by recurrence that Σ_(k=1) ^n k^4 =((n(n+1)(2n+1)(3n^2 +3n−1))/(30))

$p r o v e b y r e c u r r e n c e t h a t$ $\sum_{k = 1}^{n} k^{4} = \frac{n (n + 1) (2 n + 1) (3 n^{2} + 3 n - 1)}{30}$

Question Number 40875 Answers: 1 Comments: 0

2a sin(((25)/a)) − 51 = 0, find a

$2 a \sin (\frac{25}{a}) - 51 = 0, find a$

Question Number 40874 Answers: 0 Comments: 0

Question Number 40873 Answers: 1 Comments: 0

If a^3 +b^3 =0, prove that log (a+b)=(1/2)(log a +log b +log 3) [given a+b≠0]

$I f a^{3} + b^{3} = 0, p r o v e t h a t \log (a + b) = \frac{1}{2} (\log a + \log b + \log 3)$ $[g i v e n a + b \neq 0]$

Question Number 40872 Answers: 1 Comments: 2

If a^3 +b^3 =0, prove that log (a+b)=(1/2)(log a +log b +log 3) [given a+b≠0]

$I f a^{3} + b^{3} = 0, p r o v e t h a t \log (a + b) = \frac{1}{2} (\log a + \log b + \log 3)$ $[g i v e n a + b \neq 0]$

Question Number 40870 Answers: 1 Comments: 1

fnd ∫ (1+(1/x^2 ))arctan(x−(1/x))dx .

$f n d \int (1 + \frac{1}{x^{2}}) a r c t a n (x - \frac{1}{x}) d x .$

Question Number 40868 Answers: 0 Comments: 4

calculate ∫_0 ^(π/2) (x/(sinx))dx .

$c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{x}{s i n x} d x .$

Question Number 40867 Answers: 0 Comments: 1

Question Number 40857 Answers: 1 Comments: 2

Question Number 40847 Answers: 1 Comments: 0

If the coordinate of the points A and B be (3,3) and (7,6) then the length of the portion of the line AB intercepted between the axes is (a) (5/4) (b) ((√(10))/4) (c) ((√(13))/3) (d) none

$I f t h e c o o r d i n a t e o f t h e p o i n t s A$ $a n d B b e (3, 3) a n d (7, 6) t h e n t h e$ $l e n g t h o f t h e p o r t i o n o f t h e l i n e$ $A B i n t e r c e p t e d b e t w e e n t h e a x e s i s$ $(a) \frac{5}{4} (b) \frac{\sqrt{10}}{4} (c) \frac{\sqrt{13}}{3} (d) n o n e$

Question Number 40830 Answers: 0 Comments: 1

find ∫ (√(2+tan^2 t))dt

$f i n d \int \sqrt{2 + {t a n}^{2} t} d t$

Question Number 40829 Answers: 1 Comments: 0

let f(t) = ∫_0 ^∞ ((arctan(tx))/(x^3 +8))dx 1)find a simple form of f(t) 2)calculate ∫_0 ^∞ ((arctan(x))/(x^3 +8))dx .

$l e t f (t) = \int_{0}^{\infty} \frac{a r c t a n (t x)}{x^{3} + 8} d x$ $1) f i n d a s i m p l e f o r m o f f (t)$ $2) c a l c u l a t e \int_{0}^{\infty} \frac{a r c t a n (x)}{x^{3} + 8} d x .$

Question Number 40826 Answers: 1 Comments: 0

Question Number 40823 Answers: 0 Comments: 1

calculate ∫_0 ^(π/2) (√(cos^2 x +3sin^2 x))dx

$c a l c u l a t e \int_{0}^{\frac{π}{2}} \sqrt{{c o s}^{2} x + 3 {s i n}^{2} x} d x$

Question Number 40822 Answers: 2 Comments: 0

Question Number 40818 Answers: 1 Comments: 0

Two large parallel metal plates carrying opposite charges are seperated by a distance of 0.2m,and the potential difference between them is 0.5KV. (a)Calculate the magnitude of the uniform electric field between them. (b)Also calculate the workdone by this field on a charge of 4.0×10^(−9) C when it moves from the plate of higher potential to that of lower potential.

$T w o l a r g e p a r a l l e l m e t a l p l a t e s c a r r y i n g$ $o p p o s i t e c h a r g e s a r e s e p e r a t e d b y a$ $d i s t a n c e o f 0.2 m, a n d t h e p o t e n t i a l$ $d i f f e r e n c e b e t w e e n t h e m i s 0.5 K V .$ $(a) C a l c u l a t e t h e m a g n i t u d e o f t h e$ $u n i f o r m e l e c t r i c f i e l d b e t w e e n t h e m .$ $(b) A l s o c a l c u l a t e t h e w o r k d o n e b y$ $t h i s f i e l d o n a c h a r g e o f 4.0 \times 10^{- 9} C$ $w h e n i t m o v e s f r o m t h e p l a t e o f$ $h i g h e r p o t e n t i a l t o t h a t o f l o w e r$ $p o t e n t i a l .$

Question Number 40817 Answers: 1 Comments: 0

An electric field of 2.66KV/m,and a magnetic field of 1.20T,act on a moving electron.Calculate the speed of the electron if the resultant force due to both fields is zero.

$A n e l e c t r i c f i e l d o f 2.66 K V / m, a n d$ $a m a g n e t i c f i e l d o f 1.20 T, a c t o n a$ $m o v i n g e l e c t r o n . C a l c u l a t e t h e s p e e d$ $o f t h e e l e c t r o n i f t h e r e s u l t a n t f o r c e$ $d u e t o b o t h f i e l d s i s z e r o .$

Question Number 40816 Answers: 0 Comments: 0

An electric field of 2.66KV/m,and a magnetic field of 1.20T,act on a moving electron.Calculate the speed of the electron if the resultant force due to both fields is zero.

Question Number 40815 Answers: 1 Comments: 0

Calculate the magnetic force,F_b on an electron with velocity,v=20j^ m/s which enters a magnetic field of flux density,B=0.6k^ T

$C a l c u l a t e t h e m a g n e t i c f o r c e, F_{b} o n$ $a n e l e c t r o n w i t h v e l o c i t y, v = 20 \hat{j m} / s$ $w h i c h e n t e r s a m a g n e t i c f i e l d o f$ $f l u x d e n s i t y, B = 0.6 \hat{k T}$

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