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Question Number 33628 Answers: 0 Comments: 4

Question Number 33622 Answers: 0 Comments: 0

Question Number 33619 Answers: 1 Comments: 3

∫x^(5/2) (1−x)^(3/2) dx

$$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$

Question Number 33616 Answers: 1 Comments: 4

please help me please is there any app for practicing calculus for a CBT exam.I mean one that has a timer so I can asses my speed.

$${please}\:{help}\:{me} \\ $$$$ \\ $$$${please}\:{is}\:{there}\:{any}\:{app}\:{for}\:{practicing} \\ $$$${calculus}\:{for}\:\:{a}\:{CBT}\:{exam}.{I}\:{mean} \\ $$$${one}\:{that}\:{has}\:{a}\:{timer}\:{so}\:{I}\:{can}\:{asses} \\ $$$${my}\:{speed}. \\ $$

Question Number 33599 Answers: 1 Comments: 2

calculatef(a)= ∫_(−a) ^a (dx/((t^2 +x^2 )^(3/2) )) with a>0 .

$${calculatef}\left({a}\right)=\:\:\int_{−{a}} ^{{a}} \:\:\:\:\frac{{dx}}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\:{with}\:{a}>\mathrm{0}\:. \\ $$

Question Number 33597 Answers: 0 Comments: 0

study and give the graph of f(x) =e^(2/(lnx)) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)\:={e}^{\frac{\mathrm{2}}{{lnx}}} \:. \\ $$

Question Number 33596 Answers: 0 Comments: 0

1) prove that ∀(a,b)∈R^2 ∣sinb −sina∣≤∣b−a∣ 2)let give the sequence x_0 =0 and x_(n+1) =a +(1/2)sin(x_n ) prove that for m≥n ∣x_m −x_n ∣ ≤ ((∣a∣)/2^(n−1) ) 3) prove that (x_n ) is convergent and its limit is solution of the equation x = a +(1/2) sinx .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\left({a},{b}\right)\in{R}^{\mathrm{2}} \:\:\:\:\mid{sinb}\:−{sina}\mid\leqslant\mid{b}−{a}\mid \\ $$$$\left.\mathrm{2}\right){let}\:{give}\:{the}\:{sequence}\:\:{x}_{\mathrm{0}} =\mathrm{0}\:{and} \\ $$$${x}_{{n}+\mathrm{1}} ={a}\:+\frac{\mathrm{1}}{\mathrm{2}}{sin}\left({x}_{{n}} \right)\:{prove}\:{that}\:{for}\:{m}\geqslant{n} \\ $$$$\mid{x}_{{m}} \:−{x}_{{n}} \mid\:\leqslant\:\:\frac{\mid{a}\mid}{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{convergent}\:{and}\:{its}\:{limit}\:{is}\:{solution} \\ $$$${of}\:{the}\:{equation}\:\:{x}\:=\:{a}\:+\frac{\mathrm{1}}{\mathrm{2}}\:{sinx}\:. \\ $$

Question Number 33595 Answers: 0 Comments: 0

find lim_(x→0) ((ln(1+sinx) −x(√(1−x)))/(sinx −shx)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left(\mathrm{1}+{sinx}\right)\:−{x}\sqrt{\mathrm{1}−{x}}}{{sinx}\:−{shx}}\:\:. \\ $$

Question Number 33594 Answers: 0 Comments: 1

calculate lim_(x→1^− ) (1/((1−x)^α ))(arcsinx −(π/2)) .

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{1}^{−} } \:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\alpha} }\left({arcsinx}\:−\frac{\pi}{\mathrm{2}}\right)\:. \\ $$

Question Number 33593 Answers: 0 Comments: 0

calculate lim_(x→0) ((2(1−cosx)sinx −x^3 (1−x^2 )^(1/4) )/(sin^5 x −x^5 ))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\mathrm{2}\left(\mathrm{1}−{cosx}\right){sinx}\:−{x}^{\mathrm{3}} \:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{sin}^{\mathrm{5}} {x}\:−{x}^{\mathrm{5}} } \\ $$

Question Number 33592 Answers: 0 Comments: 0

let f(x) =e^(−x^2 ) 1) prove that f^((n)) (x) = p_n (x).e^(−x^2 ) where p_n is a polynome with deg=n 2) prove that ∀ n≥1 p_(n+1) (x) +α(x)p_n (x) +β(n)p_(n−1) (x) =0 find α and β 3)calculate p_0 ,p_1 ,p_2 ,p_3 4) calculate p_n ^(′′) (x) interms of p^′ (x) and p_n (x).

$${let}\:{f}\left({x}\right)\:={e}^{−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)\:=\:{p}_{{n}} \left({x}\right).{e}^{−{x}^{\mathrm{2}} } \:\:\:{where}\:{p}_{{n}} {is}\:{a}\:{polynome} \\ $$$${with}\:{deg}={n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\geqslant\mathrm{1}\: \\ $$$${p}_{{n}+\mathrm{1}} \left({x}\right)\:+\alpha\left({x}\right){p}_{{n}} \left({x}\right)\:+\beta\left({n}\right){p}_{{n}−\mathrm{1}} \left({x}\right)\:=\mathrm{0}\:\:{find}\:\alpha\:{and}\:\beta \\ $$$$\left.\mathrm{3}\right){calculate}\:{p}_{\mathrm{0}} ,{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{p}_{{n}} ^{''} \:\left({x}\right)\:{interms}\:{of}\:{p}^{'} \left({x}\right)\:{and}\:{p}_{{n}} \left({x}\right). \\ $$

Question Number 33591 Answers: 1 Comments: 1

find the value of Σ_(n=1) ^∞ (2/(n^3 +3n^2 +2n)) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{2}}{{n}^{\mathrm{3}} \:\:+\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{2}{n}}\:. \\ $$

Question Number 33590 Answers: 0 Comments: 1

let α >1 calculate f(α) = ∫_α ^(+∞) ((x^2 −x+1)/((x−1)^2 (x+1)^2 )) dx .

$${let}\:\alpha\:>\mathrm{1}\:\:{calculate}\:{f}\left(\alpha\right)\:=\:\int_{\alpha} ^{+\infty} \:\:\frac{{x}^{\mathrm{2}} −{x}+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 33589 Answers: 0 Comments: 1

1) decompose F(x) = (1/((x^2 +4)(x−3)^2 )) 2) calculate ∫_4 ^(+∞) (dx/((x^2 +4)(x−3)^2 )) .

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)\left({x}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{4}} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)\left({x}−\mathrm{3}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33588 Answers: 1 Comments: 1

find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)(2n+1))) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}\:. \\ $$

Question Number 33587 Answers: 0 Comments: 0

let f(x)=∫_0 ^π ln (x^2 −2x cosθ +1)dθ with ∣x∣<1 give a simple form of f(x).

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\:\left({x}^{\mathrm{2}} \:−\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${give}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 33586 Answers: 0 Comments: 0

find Σ_(n=1) ^∞ z^n ((sin(nθ))/n) with z from C and ∣z∣<1 .

$${find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{z}^{{n}} \:\:\frac{{sin}\left({n}\theta\right)}{{n}}\:\:{with}\:{z}\:{from}\:{C}\:{and}\:\mid{z}\mid<\mathrm{1}\:. \\ $$

Question Number 33585 Answers: 0 Comments: 0

study the nature of Σ_(n=1) ^∞ sin(πen!) .

$${study}\:{the}\:{nature}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} {sin}\left(\pi{en}!\right)\:. \\ $$

Question Number 33584 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ (((1+i)^n cos(nθ))/2^n ) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(\mathrm{1}+{i}\right)^{{n}} \:{cos}\left({n}\theta\right)}{\mathrm{2}^{{n}} }\:. \\ $$

Question Number 33583 Answers: 0 Comments: 0

let z∈C / ∣z∣<1 calculate Σ_(n=1) ^∞ z^n cos(nθ) and Σ_(n=1) ^∞ z^n sin(nθ)

$${let}\:{z}\in{C}\:\:/\:\mid{z}\mid<\mathrm{1}\:\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{z}^{{n}} {cos}\left({n}\theta\right) \\ $$$${and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{z}^{{n}} \:{sin}\left({n}\theta\right) \\ $$

Question Number 33582 Answers: 0 Comments: 0

find the nature of Σ_(n=1) ^∞ (n^(ln(n)) /e^(√n) ) .

$${find}\:{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}^{{ln}\left({n}\right)} }{{e}^{\sqrt{{n}}} }\:\:. \\ $$

Question Number 33581 Answers: 0 Comments: 0

find the nature of Σ_(n=1) ^∞ (−1)^n (−(1/n) +arctan((1/n)))^(1/3)

$${find}\:{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \left(−\frac{\mathrm{1}}{{n}}\:+{arctan}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$

Question Number 33575 Answers: 3 Comments: 0

simplify (√((3+2(√(2)))))

$$\mathrm{simplify}\:\sqrt{\left(\mathrm{3}+\mathrm{2}\sqrt{\left.\mathrm{2}\right)}\right.} \\ $$

Question Number 33571 Answers: 1 Comments: 0

f(x) = x^(20) + a_1 x^(19) + a_2 x^(18) + ... + a_(20) If f(1) = f(2) = f(3) = ... = f(20) What is the value of a_1 ?

$${f}\left({x}\right)\:=\:{x}^{\mathrm{20}} \:+\:{a}_{\mathrm{1}} {x}^{\mathrm{19}} \:+\:{a}_{\mathrm{2}} {x}^{\mathrm{18}} \:+\:...\:+\:{a}_{\mathrm{20}} \\ $$$$\mathrm{If}\:{f}\left(\mathrm{1}\right)\:=\:{f}\left(\mathrm{2}\right)\:=\:{f}\left(\mathrm{3}\right)\:=\:...\:=\:{f}\left(\mathrm{20}\right) \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{1}} \:? \\ $$

Question Number 33570 Answers: 1 Comments: 3

A = Σ_(n=2) ^(2017) [∫_1 ^n 2tan^(−1) x + sin^(−1) (((2x)/(1 + x^2 ))) dx] B = Π_(n=2) ^(2017) [∫_1 ^n 2tan^(−1) x + sin^(−1) (((2x)/(1 + x^2 ))) dx] A + B = ...

$${A}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\sum}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:{dx}\right] \\ $$$${B}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\prod}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:{dx}\right] \\ $$$${A}\:+\:{B}\:=\:... \\ $$

Question Number 33569 Answers: 1 Comments: 0

Given f(x) = x^3 + ax^2 + bx + c with a, b, c ∈ R, the roots are x_1 , x_2 , x_3 ∈ R Let λ is an positive integer that satisfied x_2 − x_1 = λ x_3 > (1/2)(x_1 + x_2 ) What is the max value of ((2a^3 + 27c − 9ab)/λ^3 ) ?

$$\mathrm{Given}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c} \\ $$$$\mathrm{with}\:{a},\:{b},\:{c}\:\in\:\mathbb{R},\:\mathrm{the}\:\mathrm{roots}\:\mathrm{are}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} \:\in\:\mathbb{R} \\ $$$$\mathrm{Let}\:\lambda\:\mathrm{is}\:\mathrm{an}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{that}\:\mathrm{satisfied} \\ $$$${x}_{\mathrm{2}} \:−\:{x}_{\mathrm{1}} \:=\:\lambda \\ $$$${x}_{\mathrm{3}} \:>\:\frac{\mathrm{1}}{\mathrm{2}}\left({x}_{\mathrm{1}} \:+\:{x}_{\mathrm{2}} \right) \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{max}\:\mathrm{value}\:\mathrm{of}\:\:\frac{\mathrm{2}{a}^{\mathrm{3}} \:+\:\mathrm{27}{c}\:−\:\mathrm{9}{ab}}{\lambda^{\mathrm{3}} }\:? \\ $$

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