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Question Number 32382 Answers: 2 Comments: 2

If the equation ax^2 +2bx−3c=0 has no real roots and (((3c)/4))< a+b, then

$$\mathrm{If}\:\mathrm{the}\:\mathrm{equation}\:{ax}^{\mathrm{2}} +\mathrm{2}{bx}−\mathrm{3}{c}=\mathrm{0}\:\mathrm{has} \\ $$$$\mathrm{no}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{and}\:\left(\frac{\mathrm{3}{c}}{\mathrm{4}}\right)<\:{a}+{b},\:\mathrm{then} \\ $$

Question Number 32380 Answers: 1 Comments: 2

Question Number 32379 Answers: 1 Comments: 0

Question Number 32376 Answers: 4 Comments: 1

Question Number 32369 Answers: 0 Comments: 0

prove that n^(−α) ∼ ∫_n ^(n+1) t^(−α) dt 2) prove that Σ_(k=1) ^n (1/k^α ) ∼ (n^(1−α) /(1−α)) if α<1 and Σ_(k=1) ^n (1/k^α ) ∼ ln(n) if α=1 .

$${prove}\:{that}\:\:{n}^{−\alpha} \:\sim\:\int_{{n}} ^{{n}+\mathrm{1}} \:{t}^{−\alpha} {dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:\:\frac{{n}^{\mathrm{1}−\alpha} }{\mathrm{1}−\alpha}\:{if}\:\:\alpha<\mathrm{1}\:{and} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:{ln}\left({n}\right)\:{if}\:\alpha=\mathrm{1}\:. \\ $$

Question Number 32367 Answers: 0 Comments: 0

let α∈R and x^2 ≠1 find the value of f(x) = ∫_0 ^π ln(x^2 −2x cost +1)dt calculate f(x).

$${let}\:\alpha\in{R}\:{and}\:{x}^{\mathrm{2}} \neq\mathrm{1}\:\:{find}\:{the}\:{value}\:{of} \\ $$$${f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left({x}^{\mathrm{2}} −\mathrm{2}{x}\:{cost}\:+\mathrm{1}\right){dt} \\ $$$${calculate}\:{f}\left({x}\right). \\ $$

Question Number 32365 Answers: 0 Comments: 3

let F(x) = ∫_0 ^π ln(1+xcosθ)dθ .with ∣x∣<1 find F(x) .

$${let}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\mathrm{1}+{xcos}\theta\right){d}\theta\:.{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${find}\:{F}\left({x}\right)\:. \\ $$

Question Number 32364 Answers: 0 Comments: 1

let u_n = (e −(1+(1/n))^n )^((√(n^2 +2)) −(√(n^2 +1))) find lim u_n

$${let}\:\:{u}_{{n}} =\:\left({e}\:−\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \right)^{\sqrt{{n}^{\mathrm{2}} \:+\mathrm{2}}\:\:−\sqrt{{n}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$$${find}\:\:{lim}\:{u}_{{n}} \\ $$

Question Number 32363 Answers: 0 Comments: 1

let consider the function f(x,θ) = ∫_x ^x^2 ln( 2+sinθ cost)dt calculate (∂f/∂x)(x,θ) and (∂f/∂θ)(x,θ) .

$${let}\:{consider}\:{the}\:{function} \\ $$$${f}\left({x},\theta\right)\:=\:\:\int_{{x}} ^{{x}^{\mathrm{2}} } {ln}\left(\:\mathrm{2}+{sin}\theta\:{cost}\right){dt} \\ $$$${calculate}\:\frac{\partial{f}}{\partial{x}}\left({x},\theta\right)\:{and}\:\:\frac{\partial{f}}{\partial\theta}\left({x},\theta\right)\:. \\ $$

Question Number 32362 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/((2x+1)(2x+3)(2x+5))) .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{3}\right)\left(\mathrm{2}{x}+\mathrm{5}\right)}\:. \\ $$

Question Number 32361 Answers: 0 Comments: 1

let give a>0 find ∫_0 ^∞ (e^(−x) /(√(x+a))) dx.

$${let}\:{give}\:{a}>\mathrm{0}\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{x}} }{\sqrt{{x}+{a}}}\:{dx}. \\ $$

Question Number 32360 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (dx/((2x+1)(2x+3))) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)\left(\mathrm{2}{x}+\mathrm{3}\right)}\:. \\ $$

Question Number 32359 Answers: 0 Comments: 1

find ∫_0 ^∞ (dx/((1+x^2 )(1+x^4 ))) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$

Question Number 32357 Answers: 1 Comments: 0

x^2 +ax−24=0 root is integer a range i cant speak english well. sorry

$${x}^{\mathrm{2}} +{ax}−\mathrm{24}=\mathrm{0} \\ $$$${root}\:{is}\:{integer} \\ $$$${a}\:\:\:{range} \\ $$$$ \\ $$$${i}\:{cant}\:{speak}\:{english}\:{well}.\:{sorry} \\ $$

Question Number 32354 Answers: 0 Comments: 1

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

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

Question Number 32353 Answers: 1 Comments: 0

calculate ∫_0 ^(π/4) cos(x)ln(cos(x))dx .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}\left({x}\right){ln}\left({cos}\left({x}\right)\right){dx}\:. \\ $$

Question Number 32352 Answers: 1 Comments: 2

find the value of ∫_0 ^1 arctan((√(1−x^2 )))dx

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

Question Number 32351 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) (dt/(1+cosθ sint)) .

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\:\frac{{dt}}{\mathrm{1}+{cos}\theta\:{sint}}\:.\: \\ $$

Question Number 32350 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 32349 Answers: 0 Comments: 1

find the value of ∫_0 ^π ((xdx)/(1+sinx)) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xdx}}{\mathrm{1}+{sinx}}\:. \\ $$

Question Number 32348 Answers: 0 Comments: 0

1)let n ∈Nand A_n = ∫_0 ^π (dx/(1+cos^2 (nx))) .calculate A_n 2) f∈ C^0 ([0,π], R) find lim_(n→∞) ∫_0 ^π ((f(x))/(1+cos^2 (nx)))dx .

$$\left.\mathrm{1}\right){let}\:{n}\:\in{Nand}\:\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:.{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{f}\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],\:{R}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx}\:. \\ $$

Question Number 32346 Answers: 0 Comments: 0

let u_0 =1 and u_(n+1) = u_n ((1+2u_n )/(1+3n)) give a equivalent of u_(n )

$${let}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:\:{u}_{{n}+\mathrm{1}} =\:{u}_{{n}} \:\frac{\mathrm{1}+\mathrm{2}{u}_{{n}} }{\mathrm{1}+\mathrm{3}{n}} \\ $$$${give}\:{a}\:{equivalent}\:{of}\:{u}_{{n}\:} \\ $$

Question Number 32345 Answers: 0 Comments: 0

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

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

Question Number 32344 Answers: 0 Comments: 0

let u_n = Σ_(k=1) ^n ch((1/(√(k+n)))) −n prove that u_n is convergent and find its limit.

$${let}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{ch}\left(\frac{\mathrm{1}}{\sqrt{{k}+{n}}}\right)\:−{n} \\ $$$${prove}\:{that}\:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{limit}. \\ $$

Question Number 32343 Answers: 1 Comments: 1

calculate ∫_0 ^(2π) (dt/(x−e^(it) )) .

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

Question Number 32342 Answers: 0 Comments: 0

find the value of ∫∫_D ((dxdy)/((4x^2 +y^2 +1)^2 )) D={(x,y)∈ R^2 / x^2 +y^2 ≤1 and y ≤2x } .

$${find}\:{the}\:{value}\:{of}\:\int\int_{{D}} \:\:\:\frac{{dxdy}}{\left(\mathrm{4}{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${D}=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{1}\:{and}\:{y}\:\leqslant\mathrm{2}{x}\:\right\}\:. \\ $$

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