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Question Number 34661 Answers: 0 Comments: 0

let f(x)= ∫_0 ^1 (e^(−(1+t^2 )x) /(1+t^2 )) dt find a simple form of f(x)

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

Question Number 34653 Answers: 0 Comments: 0

Question Number 34647 Answers: 2 Comments: 0

if xsin^3 θ + ycos^3 θ=sinθcosθ and xsinθ −ycosθ=0 prove that x^2 + y^2 =1

$i f {x s i n}^{3} θ + {y c o s}^{3} θ = s i n θ c o s θ$ $a n d x s i n θ - y c o s θ = 0$ $p r o v e t h a t x^{2} + y^{2} = 1$

Question Number 34646 Answers: 1 Comments: 0

show that 2tan^(−1) 2 + tan^(−1) 3= Π +tan^(−1) (1/3)

$s h o w t h a t 2 {t a n}^{- 1} 2 + {t a n}^{- 1} 3 =$ $Π + {t a n}^{- 1} \frac{1}{3}$

Question Number 34645 Answers: 0 Comments: 0

show that 2tan^(−1) 2 + tan^(−1) 3= Π +tan^(−1) (1/3)

$s h o w t h a t 2 {t a n}^{- 1} 2 + {t a n}^{- 1} 3 =$ $Π + {t a n}^{- 1} \frac{1}{3}$

Question Number 34639 Answers: 2 Comments: 0

Question Number 34635 Answers: 2 Comments: 4

calculate A(α) = ∫_0 ^1 ln(1+αix)dx 2) calculate ∫_0 ^1 ln(1+ix) dx (i^2 =−1)

$c a l c u l a t e A (α) = \int_{0}^{1} l n (1 + α i x) d x$ $2) c a l c u l a t e \int_{0}^{1} l n (1 + i x) d x (i^{2} = - 1)$

Question Number 34634 Answers: 1 Comments: 1

let f(x)=ln(1+ix) with ∣x∣<1 1) extract Re(f(x)) and Im(f(x)) 2) developp f(x) at integr serie.

$l e t f (x) = l n (1 + i x) w i t h ∣ x ∣< 1$ $1) e x t r a c t R e (f (x)) a n d I m (f (x))$ $2) d e v e l o p p f (x) a t i n t e g r s e r i e .$

Question Number 34633 Answers: 0 Comments: 0

let f(α) = ∫_(−∞) ^(+∞) ((arctan(1+αxi))/(1+x^2 ))dx find f(α) .

$l e t f (α) = \int_{- \infty}^{+ \infty} \frac{a r c t a n (1 + α x i)}{1 + x^{2}} d x$ $f i n d f (α) .$

Question Number 34632 Answers: 0 Comments: 0

let z∈C developp at integrserie f(z)=ln(1+z) with ∣z∣<1 . 2) give ln(2+i) at form of serie.

$l e t z \in C d e v e l o p p a t i n t e g r s e r i e$ $f (z) = l n (1 + z) w i t h ∣ z ∣< 1 .$ $2) g i v e l n (2 + i) a t f o r m o f s e r i e .$

Question Number 34617 Answers: 2 Comments: 2

Question Number 34615 Answers: 0 Comments: 0

decompose inside R(x) the fraction F(x)= (x^2 /((x+1)^5 ( x+3)^8 ))

$d e c o m p o s e i n s i d e R (x) t h e f r a c t i o n$ $F (x) = \frac{x^{2}}{{(x + 1)}^{5} {(x + 3)}^{8}}$

Question Number 34608 Answers: 0 Comments: 0

Question Number 34607 Answers: 0 Comments: 0

let p∈C[x] degp=n (x_i )_(1≤k≤n) the roots of p(x) a∈C?/p(a)≠0 1) calculate S_1 = Σ_(k=1) ^n (1/(x_k −a)) interms of p,p^′ and a 2)calculste S_2 =Σ_(k=1) ^n (1/((x_k −a)^2 )) interms of p,p^, p^(′′) and a.

$l e t p \in C [x] d e g p = n {(x_{i})}_{1 ⩽ k ⩽ n} t h e r o o t s o f p (x)$ $a \in C ? / p (a) \neq 0$ $1) c a l c u l a t e S_{1} = \sum_{k = 1}^{n} \frac{1}{x_{k} - a} i n t e r m s o f p, p^{^{'}} a n d a$ $2) c a l c u l s t e S_{2} = \sum_{k = 1}^{n} \frac{1}{{(x_{k} - a)}^{2}} i n t e r m s o f p, p^{,}$ $p^{^{″}} a n d a .$

Question Number 34606 Answers: 0 Comments: 0

let give p(x)=(x+1)^n −(x−1)^n 1) factorize p(x) inside C[x] 2) find the value of Π_(k=1) ^p cotan(((kπ)/(2p+1)))

$l e t g i v e p (x) = {(x + 1)}^{n} - {(x - 1)}^{n}$ $1) f a c t o r i z e p (x) i n s i d e C [x]$ $2) f i n d t h e v a l u e o f \prod_{k = 1}^{p} c o t a n (\frac{k π}{2 p + 1})$

Question Number 34605 Answers: 0 Comments: 0

decompose inside R(x) thefraction F(x)= ((x^5 +1)/(x^2^ (x−1)^2 )) .

$d e c o m p o s e i n s i d e R (x) t h e f r a c t i o n$ $F (x) = \frac{x^{5} + 1}{x^{2^{}} {(x - 1)}^{2}} .$

Question Number 34604 Answers: 0 Comments: 0

let p(x)= x^n +x+1 ∈C[x] and z∈C/p(z)=0 prove that ∣z∣<2 .

$l e t p (x) = x^{n} + x + 1 \in C [x] a n d z \in C / p (z) = 0$ $p r o v e t h a t ∣ z ∣< 2 .$

Question Number 34603 Answers: 0 Comments: 0

prove that ∀ p∈K[x] p(x) −x divide p(p(x))−x

$p r o v e t h a t \forall p \in K [x] p (x) - x d i v i d e p (p (x)) - x$

Question Number 34602 Answers: 0 Comments: 0

simplify Σ_(k=0) ^n ((k/n) −α)^2 C_n ^k x^k (1−x)^(n−k ) α∈C.

$s i m p l i f y \sum_{k = 0}^{n} {(\frac{k}{n} - α)}^{2} C_{n}^{k} x^{k} {(1 - x)}^{n - k}$ $α \in C .$

Question Number 34596 Answers: 0 Comments: 0

1) prove that Σ_(k=1) ^n H_k =(n+1)H_n −n 2) prove that Σ_(k=1) ^n H_k ^2 =(n+1)H_n ^2 −(3n+1)H_n +2n H_n =Σ_(k=1) ^n (1/k) .

$1) p r o v e t h a t \sum_{k = 1}^{n} H_{k} = (n + 1) H_{n} - n$ $2) p r o v e t h a t \sum_{k = 1}^{n} H_{k}^{2} = (n + 1) H_{n}^{2} - (3 n + 1) H_{n} + 2 n$ $H_{n} = \sum_{k = 1}^{n} \frac{1}{k} .$

Question Number 34595 Answers: 0 Comments: 0

simplify Σ_(k=1) ^n (((−1)^(k−1) )/k) C_n ^k

$s i m p l i f y \sum_{k = 1}^{n} \frac{{(- 1)}^{k - 1}}{k} C_{n}^{k}$

Question Number 34594 Answers: 0 Comments: 0

prove that Σ_(k=0) ^p (−1)^k C_n ^k =(−1)^p C_(n−1) ^p

$p r o v e t h a t \sum_{k = 0}^{p} {(- 1)}^{k} C_{n}^{k} = {(- 1)}^{p} C_{n - 1}^{p}$

Question Number 34593 Answers: 0 Comments: 0

1) calculate ∫_(−∞) ^(+∞) ((cos(αx^n ))/(x^2 +x +1)) dx with n integr natural 2) find the value of ∫_(−∞) ^∞ ((cos( α x^(2n) ))/(x^2 +x +1))dx 3) calculate ∫_(−∞) ^(+∞) ((cos(π x^3 ))/(x^2 +x +1)) dx

$1) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{c o s (α x^{n})}{x^{2} + x + 1} d x w i t h n i n t e g r$ $n a t u r a l$ $2) f i n d t h e v a l u e o f \int_{- \infty}^{\infty} \frac{c o s (α x^{2 n})}{x^{2} + x + 1} d x$ $3) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{c o s (π x^{3})}{x^{2} + x + 1} d x$

Question Number 34587 Answers: 2 Comments: 1

Question Number 34585 Answers: 1 Comments: 1

Question Number 34571 Answers: 0 Comments: 0

A machine with a velocity ratio of 5 requires 150J of work to raise a 500N load through a vertical distance of 200cm,calculate: a)the efficiency b)the M.A of the machine

$A m a c h i n e w i t h a v e l o c i t y r a t i o$ $o f 5 r e q u i r e s 150 J o f w o r k t o r a i s e$ $a 500 N l o a d t h r o u g h a v e r t i c a l$ $d i s t a n c e o f 200 c m, c a l c u l a t e :$ $a) t h e e f f i c i e n c y$ $b) t h e M . A o f t h e m a c h i n e$

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