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AllQuestion and Answers: Page 190

Question Number 196954 Answers: 1 Comments: 0

Question Number 196950 Answers: 1 Comments: 0

Prove that ∫^( (π/2)) _( 0) ((ln(1+αsint))/(sint))dt= (π^2 /8)−(1/2)(arccosα)^2

$Prove that {\int_{0}}^{\frac{π}{2}} \frac{\ln (1 + α \sin t)}{\sin t} d t = \frac{π^{2}}{8} - \frac{1}{2} {(\arccos α)}^{2}$

Question Number 196946 Answers: 2 Comments: 0

Question Number 196940 Answers: 1 Comments: 0

Question Number 196938 Answers: 1 Comments: 0

Question Number 196953 Answers: 1 Comments: 0

Question Number 196934 Answers: 1 Comments: 0

Question Number 196929 Answers: 1 Comments: 0

Question Number 196928 Answers: 0 Comments: 1

I_n =∫_0 ^(π/2) sin^n x dx

$I_{n} = \int_{0}^{\frac{π}{2}} {s i n}^{n} x d x$

Question Number 196918 Answers: 1 Comments: 0

Question Number 196917 Answers: 3 Comments: 0

Question Number 196916 Answers: 0 Comments: 0

Question Number 196915 Answers: 0 Comments: 1

Question Number 196914 Answers: 2 Comments: 0

Question Number 196913 Answers: 1 Comments: 0

soit {_(r_(n+1) =r_n /(2+r_n ^2 )) ^(r_0 =1) demontrer sans recurrence que r_n >0 demontrer par recurrence que r_(n+1) ≤(1/2)r_n demontrer sans recurrence que r_n ≤((1/2))^n •erly rolvinst•

$s o i t {_{r_{n + 1} = r_{n} / (2 + r_{n}^{2})}^{r_{0} = 1}$ $d e m o n t r e r s a n s r e c u r r e n c e q u e r_{n} > 0$ $d e m o n t r e r p a r r e c u r r e n c e q u e r_{n + 1} \leq \frac{1}{2} r_{n}$ $d e m o n t r e r s a n s r e c u r r e n c e q u e r_{n} \leq {(\frac{1}{2})}^{n}$ $∙ e r l y r o l v i n s t ∙$

Question Number 196910 Answers: 0 Comments: 0

Question Number 196902 Answers: 1 Comments: 0

Question Number 196893 Answers: 1 Comments: 0

Question Number 196886 Answers: 2 Comments: 0

Question Number 196885 Answers: 2 Comments: 0

Question Number 196883 Answers: 1 Comments: 0

Question Number 196881 Answers: 1 Comments: 1

Question Number 196880 Answers: 1 Comments: 0

1. A container of milk is (4/5) full. When 10L of milk is poured into it the container becomes (9/(10)) full. What is the capacity of the container?

$1 . A c o n t a i n e r o f m i l k i s \frac{4}{5} f u l l . W h e n 10 L o f m i l k i s p o u r e d i n t o i t t h e c o n t a i n e r b e c o m e s \frac{9}{10} f u l l . W h a t i s t h e c a p a c i t y o f t h e c o n t a i n e r ?$

Question Number 196872 Answers: 1 Comments: 0

Let ξ be a positive Root of x^2 −2023x−1 Define a sequence ϕ_i such That ϕ_0 =1 ϕ_(n+1) =⌊ϕ_n ξ⌋, find The Remainder When ϕ_(2023 ) is divided by (√ϕ_2 )

$L e t ξ b e a p o s i t i v e R o o t o f x^{2} - 2023 x - 1$ $D e f i n e a s e q u e n c e φ_{i} s u c h T h a t φ_{0} = 1$ $φ_{n + 1} = ⌊ φ_{n} ξ ⌋, f i n d T h e R e m a i n d e r W h e n φ_{2023} i s d i v i d e d b y \sqrt{φ_{2}}$

Question Number 196870 Answers: 1 Comments: 0

let b_i ∧ a_i >0 where i∈{1,2,3,...,n}& Σ_(i=1) ^n (b_i )=λ Prove that ((λ−(b_1 +b_2 ))/((b_1 +b_2 )))(a_1 +a_2 )+((λ−(b_1 +b_3 ))/((b_1 +b_3 )))(a_1 +a_3 )+....+((λ−(b_2 +b_3 ))/((b_2 +b_3 )))(a_2 +a_3 )+...((λ−(b_(n−1) +b_n ))/((b_(n−1) +b_n )))(a_(n−1) +a_n ) ≥(√(((n(n−1)(n−2)^2 )/4)×ΣΣ_(1≤i<j≤n) (a_i a_j )))

$l e t b_{i} \land a_{i} > 0 w h e r e i \in {1, 2, 3, . . ., n} & \underset{i = 1}{\sum^{n}} (b_{i}) = λ P r o v e t h a t$ $\frac{λ - (b_{1} + b_{2})}{(b_{1} + b_{2})} (a_{1} + a_{2}) + \frac{λ - (b_{1} + b_{3})}{(b_{1} + b_{3})} (a_{1} + a_{3}) + . . . . + \frac{λ - (b_{2} + b_{3})}{(b_{2} + b_{3})} (a_{2} + a_{3}) + . . . \frac{λ - (b_{n - 1} + b_{n})}{(b_{n - 1} + b_{n})} (a_{n - 1} + a_{n})$ $⩾ \sqrt{\frac{n (n - 1) {(n - 2)}^{2}}{4} \times \underset{1 ⩽ i < j ⩽ n}{Σ Σ} (a_{i} a_{j})}$

Question Number 196861 Answers: 0 Comments: 0

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