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Question Number 208367    Answers: 0   Comments: 0

1. Find the length of each of the following (a) {x : −3 < x < 7} (b) {x : 2 ≤ x ≤ 6} ∪ {−3 ≤ x ≤ −1} (c) {x : −2 ≤ x < 5} ∪ {1 < x ≤ 7} 2. Let I=(a, b). Prove that I is measurable and m(I) = L(I).

$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following} \\ $$$$\:\:\:\:\:\left(\mathrm{a}\right)\:\left\{\mathrm{x}\::\:−\mathrm{3}\:<\:\mathrm{x}\:<\:\mathrm{7}\right\} \\ $$$$\:\:\:\:\:\left(\mathrm{b}\right)\:\left\{\mathrm{x}\::\:\mathrm{2}\:\leqslant\:\mathrm{x}\:\leqslant\:\mathrm{6}\right\}\:\cup\:\left\{−\mathrm{3}\:\leqslant\:\mathrm{x}\:\leqslant\:−\mathrm{1}\right\} \\ $$$$\:\:\:\:\:\left(\mathrm{c}\right)\:\left\{\mathrm{x}\::\:−\mathrm{2}\:\leqslant\:\mathrm{x}\:<\:\mathrm{5}\right\}\:\cup\:\left\{\mathrm{1}\:<\:\mathrm{x}\:\leqslant\:\mathrm{7}\right\} \\ $$$$ \\ $$$$\mathrm{2}.\:\mathrm{Let}\:\mathrm{I}=\left(\mathrm{a},\:\mathrm{b}\right).\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{I}\:\mathrm{is}\:\mathrm{measurable} \\ $$$$\mathrm{and}\:\mathrm{m}\left(\mathrm{I}\right)\:=\:\mathrm{L}\left(\mathrm{I}\right). \\ $$

Question Number 208362    Answers: 2   Comments: 0

P(x) is polynomial P(x) = ((x^4 + 2ax^3 − bx − 5)/((x + 1)^2 )) Find: b = ?

$$\mathrm{P}\left(\mathrm{x}\right)\:\:\mathrm{is}\:\mathrm{polynomial} \\ $$$$\mathrm{P}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{2ax}^{\mathrm{3}} \:−\:\mathrm{bx}\:−\:\mathrm{5}}{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{b}}\:=\:? \\ $$

Question Number 208361    Answers: 0   Comments: 0

Question Number 208359    Answers: 0   Comments: 1

Question Number 208354    Answers: 0   Comments: 1

Question Number 208342    Answers: 2   Comments: 0

a,b,c∈N x = 4(2a+5) = 6(b+9) = 9(c−1) find: min(x+a+b+c) = ?

$$\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N} \\ $$$$\mathrm{x}\:=\:\mathrm{4}\left(\mathrm{2a}+\mathrm{5}\right)\:=\:\mathrm{6}\left(\mathrm{b}+\mathrm{9}\right)\:=\:\mathrm{9}\left(\mathrm{c}−\mathrm{1}\right) \\ $$$$\mathrm{find}:\:\:\:\boldsymbol{\mathrm{min}}\left(\mathrm{x}+\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\:=\:? \\ $$

Question Number 208344    Answers: 4   Comments: 3

Question Number 208338    Answers: 0   Comments: 0

Question Number 208335    Answers: 1   Comments: 0

∫_(−1) ^1 (√(1−t^4 ))dt

$$\int_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{t}^{\mathrm{4}} }{dt} \\ $$

Question Number 208334    Answers: 0   Comments: 1

∫_0 ^(4/π) ln(cosx)dx

$$\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left({cosx}\right){dx} \\ $$

Question Number 208332    Answers: 1   Comments: 0

Question Number 208328    Answers: 0   Comments: 0

Question Number 208327    Answers: 1   Comments: 0

Question Number 208322    Answers: 2   Comments: 0

Question Number 208318    Answers: 1   Comments: 0

calcul lim n→+∞ ∫_0 ^(+∞) ((cos(nx))/((nx+1)(1+x^2 ) ))dx

$${calcul}\:\:\:{lim}\:{n}\rightarrow+\infty \\ $$$$\int_{\mathrm{0}} ^{+\infty} \:\frac{{cos}\left({nx}\right)}{\left({nx}+\mathrm{1}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:}{dx} \\ $$

Question Number 208316    Answers: 1   Comments: 0

∫ ((x^2 + 3)/(x^2 (x + 1)(x^2 + 1)^2 )) dx

$$\int\:\frac{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{3}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}\:\:+\:\:\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 208312    Answers: 1   Comments: 0

lim_(x→0 ((a^x −1)/x) = log a)

$${lim}_{{x}\rightarrow\mathrm{0}\:\frac{{a}^{{x}} −\mathrm{1}}{{x}}\:=\:{log}\:{a}} \\ $$

Question Number 208306    Answers: 2   Comments: 1

calcul / lim n→+∞ ∫_0 ^(+∞) f_n (x) f_n (x)= arctan((x/n))e^(−x) dx

$${calcul}\:/\:{lim}\:{n}\rightarrow+\infty\:\int_{\mathrm{0}} ^{+\infty} \:{f}_{{n}} \left({x}\right) \\ $$$$\:{f}_{{n}} \left({x}\right)=\:{arctan}\left(\frac{{x}}{{n}}\right){e}^{−{x}} {dx} \\ $$

Question Number 208303    Answers: 1   Comments: 0

Resolver (∂^2 u/∂y^2 ) − x^2 u = xe^(4y)

$${Resolver} \\ $$$$\frac{\partial^{\mathrm{2}} {u}}{\partial{y}^{\mathrm{2}} }\:−\:{x}^{\mathrm{2}} {u}\:=\:{xe}^{\mathrm{4}{y}} \\ $$

Question Number 208293    Answers: 1   Comments: 0

W Z_( ∩ + (( determinant ()2)/))

Question Number 208292    Answers: 1   Comments: 0

let T be a n×n matrix with integral entries and Q = T + (1/2)I where I denote the n×n identity matrix then prove that matrix Q is invertible

$$\:\mathrm{let}\:\mathrm{T}\:\mathrm{be}\:\mathrm{a}\:{n}×{n}\:\mathrm{matrix}\:\mathrm{with}\:\mathrm{integral}\: \\ $$$$\:\mathrm{entries}\:\mathrm{and}\:\:\mathrm{Q}\:=\:\mathrm{T}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{I}\:\:\:\mathrm{where}\:\mathrm{I}\:\mathrm{denote} \\ $$$$\:\:\mathrm{the}\:\mathrm{n}×\mathrm{n}\:\mathrm{identity}\:\mathrm{matrix}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\:\:\mathrm{that}\:\mathrm{matrix}\:\mathrm{Q}\:\mathrm{is}\:\mathrm{invertible} \\ $$

Question Number 208288    Answers: 2   Comments: 0

Question Number 208282    Answers: 1   Comments: 2

If cosα−cosβ = (1/5) sinα + sinβ = (1/2) Find cos(α + β) = ?

$$\mathrm{If}\:\:\:\mathrm{cos}\alpha−\mathrm{cos}\beta\:=\:\frac{\mathrm{1}}{\mathrm{5}}\:\mathrm{sin}\alpha\:+\:\mathrm{sin}\beta\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{Find}\:\:\:\mathrm{cos}\left(\alpha\:+\:\beta\right)\:=\:? \\ $$

Question Number 208280    Answers: 1   Comments: 0

L=∫_0 ^(4/π) ln(cosx)dx

$${L}=\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left({cosx}\right){dx} \\ $$

Question Number 208277    Answers: 2   Comments: 0

Question Number 208269    Answers: 2   Comments: 1

Question Number 208264    Answers: 1   Comments: 0

$$\:\:\:\cancel{\underline{\underbrace{ }}} \\ $$

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