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Question Number 153949 Answers: 1 Comments: 0

∫_0 ^( ∞) sin(x^2 )cos(x^3 )dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left({x}^{\mathrm{3}} \right){dx} \\ $$$$\: \\ $$

Question Number 154089 Answers: 2 Comments: 0

Question Number 154090 Answers: 0 Comments: 0

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Question Number 153946 Answers: 0 Comments: 0

show whether ∫_0 ^( ∞) sin(x^2 )cos((√x))dx is solvable

$$\: \\ $$$$\:\:\mathrm{show}\:\mathrm{whether} \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\left({x}^{\mathrm{2}} \right)\mathrm{cos}\left(\sqrt{{x}}\right){dx} \\ $$$$\:\:\mathrm{is}\:\mathrm{solvable} \\ $$$$\: \\ $$

Question Number 153934 Answers: 0 Comments: 0

Question Number 153977 Answers: 1 Comments: 0

Question Number 153924 Answers: 2 Comments: 0

∫ ((1+sin x)/(sin x(1+cos x))) dx =?

$$\int\:\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{sin}\:{x}\left(\mathrm{1}+\mathrm{cos}\:{x}\right)}\:{dx}\:=? \\ $$

Question Number 153920 Answers: 0 Comments: 0

Question Number 153918 Answers: 1 Comments: 0

the base of an object is in the form of a circle with radius 1. suppose that all section of the object are perpendicular to a diameter of a square. determine the volume of the object?

$$\mathrm{the}\:\mathrm{base}\:\mathrm{of}\:\mathrm{an}\:\mathrm{object}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with} \\ $$$$\mathrm{radius}\:\mathrm{1}.\:\mathrm{suppose}\:\mathrm{that}\:\mathrm{all}\:\mathrm{section}\:\mathrm{of}\:\mathrm{the}\:\mathrm{object}\:\mathrm{are} \\ $$$$\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}.\:\mathrm{determine} \\ $$$$\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{object}? \\ $$

Question Number 153916 Answers: 0 Comments: 0

The value of Σ_(n=0) ^∞ (((3_n )(2_n )x^n )/((1_n )n!)) β(2,n+1) is a. (1/2)Σ_(n=0) ^∞ (2_n )(x^n /(n!)) b. (1/2)Σ_(n=0) ^∞ (((3_n )(2_n ))/((1_n ))) (x^n /(n!)) c. (1/2)Σ_(n=0) ^∞ (((2_n )x^n )/((1_n )n!)) d. (1/3)Σ_(n=0) ^∞ (((3_n )x^n )/((1_n )n!))

$${The}\:{value}\:{of}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(\mathrm{3}_{{n}} \right)\left(\mathrm{2}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!}\:\beta\left(\mathrm{2},{n}+\mathrm{1}\right)\:{is} \\ $$$${a}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\mathrm{2}_{{n}} \right)\frac{{x}^{{n}} }{{n}!} \\ $$$${b}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{3}_{{n}} \right)\left(\mathrm{2}_{{n}} \right)}{\left(\mathrm{1}_{{n}} \right)}\:\frac{{x}^{{n}} }{{n}!} \\ $$$${c}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{2}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!} \\ $$$${d}.\:\frac{\mathrm{1}}{\mathrm{3}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{3}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!} \\ $$

Question Number 153915 Answers: 1 Comments: 0

Question Number 153912 Answers: 2 Comments: 0

Question Number 153905 Answers: 2 Comments: 1

Question Number 153903 Answers: 0 Comments: 0

Question Number 153901 Answers: 2 Comments: 1

Question Number 153899 Answers: 0 Comments: 0

Determine whether there exists 2016 distinct prime numbers p_1 ,p_2 ,...,p_(2016) and positive integer n such that: Σ_(i=1) ^(2016) (1/(p_i ^2 + 1)) = (1/n^2 )

$$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{there}\:\mathrm{exists}\:\:\mathrm{2016} \\ $$$$\mathrm{distinct}\:\mathrm{prime}\:\mathrm{numbers}\:\:\mathrm{p}_{\mathrm{1}} ,\mathrm{p}_{\mathrm{2}} ,...,\mathrm{p}_{\mathrm{2016}} \\ $$$$\mathrm{and}\:\mathrm{positive}\:\mathrm{integer}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\mathrm{2016}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{p}_{\boldsymbol{\mathrm{i}}} ^{\mathrm{2}} \:+\:\mathrm{1}}\:=\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 153898 Answers: 1 Comments: 0

Find all functions f:Q→Q satisfying these followong conditions for all x∈Q 1. f(x + 1) = f(x) + 1 2. f(x^3 ) = f^( 3) (x)

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{functions}\:\:\mathrm{f}:\mathrm{Q}\rightarrow\mathrm{Q}\:\:\mathrm{satisfying} \\ $$$$\mathrm{these}\:\mathrm{followong}\:\mathrm{conditions}\:\mathrm{for}\:\mathrm{all}\:\boldsymbol{\mathrm{x}}\in\mathrm{Q} \\ $$$$\mathrm{1}.\:\mathrm{f}\left(\mathrm{x}\:+\:\mathrm{1}\right)\:=\:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{1} \\ $$$$\mathrm{2}.\:\mathrm{f}\left(\mathrm{x}^{\mathrm{3}} \right)\:=\:\mathrm{f}^{\:\mathrm{3}} \left(\mathrm{x}\right) \\ $$

Question Number 153896 Answers: 1 Comments: 0

Question Number 153895 Answers: 0 Comments: 0

Question Number 153897 Answers: 0 Comments: 1

Denote x_n is the unique positive root of the following equation: x^n + x^(n−1) + ... x = n + 2 Prove that the sequence (x_n ) converges to a positive real number. Find that limit.

$$\mathrm{Denote}\:\:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \:\:\mathrm{is}\:\mathrm{the}\:\mathrm{unique}\:\mathrm{positive}\:\mathrm{root} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation}: \\ $$$$\mathrm{x}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{x}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \:+\:...\:\mathrm{x}\:=\:\mathrm{n}\:+\:\mathrm{2} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left(\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)\:\mathrm{converges} \\ $$$$\mathrm{to}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{Find}\:\mathrm{that} \\ $$$$\mathrm{limit}. \\ $$

Question Number 153893 Answers: 0 Comments: 0

∫_( 0) ^( ∞) a Π_(p → 1) ^∞ (((p^2 − x^(2n) )/p^2 ))dx, 1 < 2n < n + 1

$$\int_{\:\mathrm{0}} ^{\:\:\infty} \mathrm{a}\:\underset{\mathrm{p}\:\rightarrow\:\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{p}^{\mathrm{2}} \:\:−\:\:\:\mathrm{x}^{\mathrm{2n}} }{\mathrm{p}^{\mathrm{2}} }\right)\mathrm{dx},\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:<\:\:\mathrm{2n}\:\:<\:\:\mathrm{n}\:\:+\:\:\mathrm{1} \\ $$

Question Number 153877 Answers: 0 Comments: 1

Question Number 155421 Answers: 3 Comments: 0

Question Number 153875 Answers: 0 Comments: 0

Prove that.. 𝛗 : =∫_( 1) ^( +∞) (( ln (x ))/(( x^( π) −1 )( ln^( 2) (x) +1 )^2 ))dx= ((π^( 2) − 8)/(16)) ■

$$ \\ $$$$\:\:\:\:\mathrm{Prove}\:\:\mathrm{that}.. \\ $$$$\:\:\: \\ $$$$\:\:\:\:\boldsymbol{\phi}\::\:=\int_{\:\mathrm{1}} ^{\:+\infty} \frac{\:{ln}\:\left({x}\:\right)}{\left(\:{x}^{\:\pi} \:−\mathrm{1}\:\right)\left(\:{ln}^{\:\mathrm{2}} \left({x}\right)\:+\mathrm{1}\:\right)^{\mathrm{2}} }{dx}=\:\frac{\pi^{\:\mathrm{2}} −\:\mathrm{8}}{\mathrm{16}}\:\:\:\:\:\:\:\:\:\blacksquare\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 153873 Answers: 0 Comments: 1

∫_0 ^( ∞) (( x)/((1 +x^( 2) ) ( e^( 2πx) − 1))) dx =((2γ− 1)/4)

$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}}{\left(\mathrm{1}\:+{x}^{\:\mathrm{2}} \right)\:\left(\:{e}^{\:\mathrm{2}\pi{x}} −\:\mathrm{1}\right)}\:{dx}\:=\frac{\mathrm{2}\gamma−\:\mathrm{1}}{\mathrm{4}} \\ $$$$ \\ $$

Question Number 153870 Answers: 1 Comments: 3

find the minimum and maximum value of (5/(f(θ)+3)) where f(θ)=8cos θ−15 sin θ

$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{5}}{{f}\left(\theta\right)+\mathrm{3}}\:\mathrm{where}\:{f}\left(\theta\right)=\mathrm{8cos}\:\theta−\mathrm{15}\:\mathrm{sin}\:\theta \\ $$

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