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AllQuestion and Answers: Page 1706

Question Number 36986 Answers: 0 Comments: 0

Question Number 36985 Answers: 0 Comments: 0

Question Number 36978 Answers: 1 Comments: 1

if (a^2 /(b+c)) = (b^2 /(c+a)) = (c^2 /(a+b)) = 1 then find the value of (1/(1+a)) + (1/(1+b)) + (1/(1+c))

$${if}\:\:\frac{{a}^{\mathrm{2}} }{{b}+{c}}\:=\:\frac{{b}^{\mathrm{2}} }{{c}+{a}}\:=\:\frac{{c}^{\mathrm{2}} }{{a}+{b}}\:=\:\mathrm{1}\:{then}\:{find}\:{the}\:{value}\:{of}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{1}+{a}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+{b}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+{c}} \\ $$

Question Number 36965 Answers: 2 Comments: 3

∫((x^5 −x^4 +x^3 −1)/((x^2 −x+1)^3 ))dx=

$$\int\frac{{x}^{\mathrm{5}} −{x}^{\mathrm{4}} +{x}^{\mathrm{3}} −\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}= \\ $$

Question Number 36957 Answers: 2 Comments: 0

Interval in which given function is decreasing. f(x)= (2^x −1)(2^x −2)^2

$$\mathrm{Interval}\:\mathrm{in}\:\mathrm{which}\:\mathrm{given}\:\mathrm{function}\:\mathrm{is} \\ $$$${decreasing}. \\ $$$$\mathrm{f}\left({x}\right)=\:\left(\mathrm{2}^{{x}} −\mathrm{1}\right)\left(\mathrm{2}^{{x}} −\mathrm{2}\right)^{\mathrm{2}} \\ $$

Question Number 36953 Answers: 1 Comments: 3

lim_(n→∞) nsin (2π(√(1+n^2 )) ) ,( n∈N).

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{nsin}\:\left(\mathrm{2}\pi\sqrt{\mathrm{1}+\mathrm{n}^{\mathrm{2}} }\:\right)\:,\left(\:\mathrm{n}\in\mathbb{N}\right). \\ $$

Question Number 36948 Answers: 0 Comments: 2

calculate ∫∫_D (√(xy)) dxdy with D={(x,y)∈R^2 / (x+y)^2 ≥2x and xy≥0}

$${calculate}\:\:\:\int\int_{{D}} \sqrt{{xy}}\:{dxdy}\:\:{with} \\ $$$${D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\:\left({x}+{y}\right)^{\mathrm{2}} \:\geqslant\mathrm{2}{x}\:\:{and}\:{xy}\geqslant\mathrm{0}\right\} \\ $$

Question Number 36947 Answers: 1 Comments: 1

integrate the d.equation xy^′ +y = ((2x)/(√(1−x^4 ))) .

$${integrate}\:{the}\:{d}.{equation}\:\:{xy}^{'} \:+{y}\:=\:\frac{\mathrm{2}{x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:. \\ $$

Question Number 36946 Answers: 0 Comments: 2

calculateϕ(λ)= ∫_0 ^π ((cos(t))/(1−2λ cost +λ^2 )) dt

$${calculate}\varphi\left(\lambda\right)=\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\:\frac{{cos}\left({t}\right)}{\mathrm{1}−\mathrm{2}\lambda\:{cost}\:+\lambda^{\mathrm{2}} }\:{dt} \\ $$

Question Number 36945 Answers: 0 Comments: 0

calulate ∫_0 ^(π/4) (dx/(√(tan(x)(1−tanx))))

$${calulate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{dx}}{\sqrt{{tan}\left({x}\right)\left(\mathrm{1}−{tanx}\right)}} \\ $$

Question Number 36944 Answers: 1 Comments: 1

find ϕ(a) = ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0

$${find}\:\varphi\left({a}\right)\:=\:\int_{{a}} ^{+\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:−{a}^{\mathrm{2}} }}\:\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 36943 Answers: 0 Comments: 1

find the value of ∫_0 ^1 ((lnx)/((√x)(1−x)^(3/2) ))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{lnx}}{\sqrt{{x}}\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{dx} \\ $$

Question Number 36942 Answers: 1 Comments: 0

let I = ∫_0 ^(π/2) ((cosx)/(√(1+cosx sinx)))dx and J =∫_0 ^(π/2) ((sinx)/(√(1+cosx sinx)))dx prove that I=J then calculate I and J .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cosx}}{\sqrt{\mathrm{1}+{cosx}\:{sinx}}}{dx}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{sinx}}{\sqrt{\mathrm{1}+{cosx}\:{sinx}}}{dx} \\ $$$${prove}\:{that}\:{I}={J}\:\:{then}\:{calculate}\:{I}\:{and}\:{J}\:. \\ $$

Question Number 36941 Answers: 0 Comments: 1

calculate ∫_0 ^((3π)/4) (dt/((1+sin^2 t)^2 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\mathrm{3}\pi}{\mathrm{4}}} \:\:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{sin}^{\mathrm{2}} {t}\right)^{\mathrm{2}} } \\ $$

Question Number 36940 Answers: 0 Comments: 1

find f(a)= ∫_0 ^a arctan((√(a^2 −x^2 )))dx

$${find}\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{{a}} \:{arctan}\left(\sqrt{{a}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 36939 Answers: 0 Comments: 0

calculate ∫_0 ^1 ^3 (√(x^2 (1−x) )) dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:^{\mathrm{3}} \sqrt{{x}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)\:}\:{dx} \\ $$

Question Number 36938 Answers: 0 Comments: 2

1) find f(a) = ∫_0 ^(π/2) (dt/(1+a cost)) 2) find A(θ) =∫_0 ^(π/2) (dt/(1+sinθ cost))

$$\left.\mathrm{1}\right)\:{find}\:\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{cost}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{sin}\theta\:{cost}} \\ $$

Question Number 36937 Answers: 0 Comments: 1

calculate ∫_0 ^π ((x dx)/(1+cosx))

$${calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}\:{dx}}{\mathrm{1}+{cosx}} \\ $$

Question Number 36936 Answers: 0 Comments: 2

calculate I_n = ∫_0 ^π (dx/(1+cos^2 (nx)))

$${calculate}\:{I}_{{n}} \:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)} \\ $$

Question Number 36935 Answers: 0 Comments: 0

find all function f R→R wich verify ∀(x,y)∈ R^2 f(x).f(y) =∫_(x−y) ^(x+y) f(t)dt .

$${find}\:{all}\:{function}\:{f}\:{R}\rightarrow{R}\:\:{wich}\:{verify} \\ $$$$\forall\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:\:\:{f}\left({x}\right).{f}\left({y}\right)\:=\int_{{x}−{y}} ^{{x}+{y}} \:{f}\left({t}\right){dt}\:. \\ $$

Question Number 36934 Answers: 0 Comments: 0

calculate [ Σ_(k=1) ^(10^4 ) (1/(√k)) ].

$${calculate}\:\left[\:\sum_{{k}=\mathrm{1}} ^{\mathrm{10}^{\mathrm{4}} } \:\:\frac{\mathrm{1}}{\sqrt{{k}}}\:\right]. \\ $$

Question Number 36933 Answers: 0 Comments: 0

find lim_(n→+∞) Σ_(i=1) ^n Σ_(j=1) ^n (((−1)^(i+j) )/(i+j)) .

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\sum_{{i}=\mathrm{1}} ^{{n}} \:\sum_{{j}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{i}+{j}} }{{i}+{j}}\:. \\ $$

Question Number 36932 Answers: 0 Comments: 0

let f ∈ C^0 ([0,π],R) prove that lim_(n→+∞) ∫_0 ^π f(x) ∣sin(nx)∣dx =(2/π) ∫_0 ^π f(x)dx .

$${let}\:{f}\:\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],{R}\right)\:\:{prove}\:{that} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right)\:\mid{sin}\left({nx}\right)\mid{dx}\:=\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}\:. \\ $$

Question Number 36931 Answers: 0 Comments: 2

calculate ∫_0 ^(2π) (dt/(x −e^(it) ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{x}\:−{e}^{{it}} } \\ $$

Question Number 36930 Answers: 0 Comments: 0

let u_n = (1/(2n+1)) +(1/(2n+3)) +.....+(1/(4n−1)) calculate lim_(n→+∞) u_n .

$${let}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{3}}\:+.....+\frac{\mathrm{1}}{\mathrm{4}{n}−\mathrm{1}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} . \\ $$

Question Number 36929 Answers: 0 Comments: 0

study and give th graph of the function f(x)=x(1−(1/x))^(x+1) .

$${study}\:{and}\:{give}\:{th}\:{graph}\:{of}\:{the}\:{function} \\ $$$${f}\left({x}\right)={x}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)^{{x}+\mathrm{1}} . \\ $$

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