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Question Number 42435    Answers: 1   Comments: 1

find ∫_0 ^1 (dx/((√x) +(√(1−x))))

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\sqrt{{x}}\:+\sqrt{\mathrm{1}−{x}}} \\ $$

Question Number 42430    Answers: 1   Comments: 2

find ∫ (dx/(3+tan^2 x))

$${find}\:\int\:\:\:\:\:\:\:\frac{{dx}}{\mathrm{3}+{tan}^{\mathrm{2}} {x}} \\ $$

Question Number 42422    Answers: 1   Comments: 2

Question Number 42408    Answers: 0   Comments: 2

(√(a−(√(a+x ))))+ (√(a+(√(a−x)) )) = 2x Solve for x in terms of a

$$\sqrt{{a}−\sqrt{{a}+{x}\:}}+\:\sqrt{{a}+\sqrt{{a}−{x}}\:}\:=\:\mathrm{2}{x} \\ $$$${Solve}\:{for}\:{x}\:{in}\:{terms}\:{of}\:{a} \\ $$

Question Number 42407    Answers: 1   Comments: 1

∫ (1/(1 + tanx)) dx

$$\int\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{tanx}}\:\mathrm{dx} \\ $$

Question Number 42402    Answers: 0   Comments: 0

calculate ∫∫_(x≤x^2 +y^2 ≤1) ((dxdy)/((1+x^2 +y^2 )^2 ))

$${calculate}\:\int\int_{{x}\leqslant{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}} \:\:\:\frac{{dxdy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$

Question Number 42395    Answers: 0   Comments: 1

calculate ∫∫_D ((xy)/((1+x^2 +y^2 )))dxdy with D ={(x,y)∈ R^2 / 0≤x≤1 ,0≤y≤1, x^2 +y^2 ≤1}

$${calculate}\:\int\int_{{D}} \:\:\:\:\:\:\frac{{xy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right)}{dxdy}\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:\:/\:\:\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:,\mathrm{0}\leqslant{y}\leqslant\mathrm{1},\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{1}\right\} \\ $$

Question Number 42394    Answers: 1   Comments: 1

find ∫_0 ^1 (dt/(t+(√(1−t^2 )))) dt

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{t}+\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}\:{dt} \\ $$

Question Number 42392    Answers: 1   Comments: 1

calculate ∫ ((lnx)/(x +x(lnx)^2 ))dx

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

Question Number 42391    Answers: 1   Comments: 1

find the value of ∫_0 ^(π/4) ln(1+tanx)dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}+{tanx}\right){dx} \\ $$

Question Number 42375    Answers: 1   Comments: 5

Find value of α such that the following system has infinite many solutions x − 3z = −3 −2x − αy + z = 2 x + 2y + αz = 1

$$\mathrm{Find}\:\mathrm{value}\:\mathrm{of}\:\:\alpha\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{system} \\ $$$$\mathrm{has}\:\mathrm{infinite}\:\mathrm{many}\:\mathrm{solutions} \\ $$$$ \\ $$$${x}\:−\:\mathrm{3}{z}\:=\:−\mathrm{3} \\ $$$$−\mathrm{2}{x}\:−\:\alpha{y}\:+\:{z}\:=\:\mathrm{2} \\ $$$${x}\:+\:\mathrm{2}{y}\:+\:\alpha{z}\:=\:\mathrm{1} \\ $$

Question Number 42374    Answers: 0   Comments: 1

let f(x) = ∫_0 ^∞ arctan(xt^2 )cos(t^2 ) dt 1) find a explicite form of f^′ (x) 2) find a explicite form of f(x) 3) find the value of ∫_0 ^∞ cos(t^2 ) arctan(t^2 )dt and ∫_0 ^∞ cos(t^2 )arctan(2t^2 )dt

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:{arctan}\left({xt}^{\mathrm{2}} \right){cos}\left({t}^{\mathrm{2}} \right)\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:{cos}\left({t}^{\mathrm{2}} \right)\:{arctan}\left({t}^{\mathrm{2}} \right){dt}\:\:\:{and}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({t}^{\mathrm{2}} \right){arctan}\left(\mathrm{2}{t}^{\mathrm{2}} \right){dt} \\ $$

Question Number 42370    Answers: 1   Comments: 1

Question Number 42367    Answers: 1   Comments: 0

lim_(n→∞) ((( ((n),(0) ) ((n),(1) ) ((n),(2) )... ((n),(n) )))^(1/(n^2 +n)) )

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt[{{n}^{\mathrm{2}} +{n}}]{\:\begin{pmatrix}{{n}}\\{\mathrm{0}}\end{pmatrix}\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}...\begin{pmatrix}{{n}}\\{{n}}\end{pmatrix}}\right) \\ $$

Question Number 42366    Answers: 1   Comments: 0

∫_(2005) ^(2017) (((ln ∣x − 2017∣)^(2017) )/((ln ∣x − 2015∣)^(2017) + (ln ∣x − 2017∣)^(2017) )) dx

$$\underset{\mathrm{2005}} {\overset{\mathrm{2017}} {\int}}\:\frac{\left(\mathrm{ln}\:\mid{x}\:−\:\mathrm{2017}\mid\right)^{\mathrm{2017}} }{\left(\mathrm{ln}\:\mid{x}\:−\:\mathrm{2015}\mid\right)^{\mathrm{2017}} \:+\:\left(\mathrm{ln}\:\mid{x}\:−\:\mathrm{2017}\mid\right)^{\mathrm{2017}} }\:{dx} \\ $$

Question Number 42364    Answers: 0   Comments: 3

∫ ((x + sinx − cosx − 1)/(x + e^x + sinx)) dx

$$\int\:\:\frac{\mathrm{x}\:+\:\mathrm{sinx}\:−\:\mathrm{cosx}\:−\:\mathrm{1}}{\mathrm{x}\:+\:\mathrm{e}^{\mathrm{x}} \:+\:\mathrm{sinx}}\:\mathrm{dx} \\ $$

Question Number 42401    Answers: 0   Comments: 1

calculate Σ_(n=0) ^∞ (((−1)^n )/((n+1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$

Question Number 42400    Answers: 0   Comments: 0

find lim_(n→+∞) (1/(√n)) Σ_(k=1) ^n (1/((√k) +(√(n−k))))

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{\sqrt{{n}}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\sqrt{{k}}\:+\sqrt{{n}−{k}}} \\ $$

Question Number 42358    Answers: 1   Comments: 0

∫_( −1) ^( 1) (x^(2015) /(((1 + x))^(1/(2015)) + ((1 − x))^(1/(2015)) )) dx

$$\int_{\:−\mathrm{1}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}^{\mathrm{2015}} }{\sqrt[{\mathrm{2015}}]{\mathrm{1}\:+\:\mathrm{x}}\:\:+\:\:\sqrt[{\mathrm{2015}}]{\mathrm{1}\:−\:\mathrm{x}}\:}\:\:\mathrm{dx} \\ $$

Question Number 42357    Answers: 1   Comments: 0

Question Number 42352    Answers: 1   Comments: 0

Question Number 42345    Answers: 1   Comments: 2

tan 15° =

$$\mathrm{tan}\:\mathrm{15}°\:= \\ $$

Question Number 42336    Answers: 0   Comments: 0

find ∫ ln(x−cosx)dx .

$${find}\:\:\int\:{ln}\left({x}−{cosx}\right){dx}\:. \\ $$

Question Number 42332    Answers: 0   Comments: 0

Question Number 42331    Answers: 1   Comments: 0

Question Number 42330    Answers: 0   Comments: 0

A linear function f(x)=ax + b transforms X={1,2,3,5,9,11} into Y,so that f(5)=13 and f(1)=5 Calculate the mean and Variance of X and Y.

$${A}\:{linear}\:{function}\:{f}\left({x}\right)={ax}\:+\:{b}\:{transforms}\:{X}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{9},\mathrm{11}\right\} \\ $$$${into}\:{Y},{so}\:{that}\:{f}\left(\mathrm{5}\right)=\mathrm{13}\:{and}\:{f}\left(\mathrm{1}\right)=\mathrm{5} \\ $$$${Calculate}\:{the}\:{mean}\:{and}\:{Variance}\:{of}\:{X}\:{and}\:{Y}. \\ $$

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