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Question Number 118513 Answers: 2 Comments: 0

If f(x) + f(((x − 1)/x)) = 1 + x, find f(2).

$$\mathrm{If}\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:\:+\:\:\mathrm{f}\left(\frac{\mathrm{x}\:\:−\:\:\mathrm{1}}{\mathrm{x}}\right)\:\:\:=\:\:\:\mathrm{1}\:\:+\:\:\mathrm{x},\:\:\:\:\:\:\:\mathrm{find}\:\:\:\mathrm{f}\left(\mathrm{2}\right). \\ $$

Question Number 121164 Answers: 0 Comments: 0

...ADVANCED CALCULUS... If ∫_0 ^( ∞) ln(x)sin(x^2 )dx =λ∫_0 ^( ∞) sin(x^2 )dx then find the value of ′′λ′′ . ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\mathrm{ADVANCED}\:\:\mathrm{CALCULUS}... \\ $$$$\:\:\:\:\mathrm{If}\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left({x}\right){sin}\left({x}^{\mathrm{2}} \right){dx}\:=\lambda\int_{\mathrm{0}} ^{\:\infty} {sin}\left({x}^{\mathrm{2}} \right){dx}\: \\ $$$$\:\:\:\:\:\:\:\:{then}\:\:{find}\:\:{the}\:\:{value}\:{of}\:''\lambda''\:. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\mathrm{m}.\mathrm{n}.\mathrm{july}.\mathrm{1970}... \\ $$

Question Number 118506 Answers: 2 Comments: 0

Question Number 118511 Answers: 3 Comments: 0

How many positive integers x satisfy log_(x/8) (x^2 /4)<7+log_2 (8/x)

$$\mathrm{How}\:\mathrm{many}\:\mathrm{positive}\:\mathrm{integers}\:{x}\:\mathrm{satisfy} \\ $$$$\mathrm{log}_{\frac{{x}}{\mathrm{8}}} \frac{{x}^{\mathrm{2}} }{\mathrm{4}}<\mathrm{7}+\mathrm{log}_{\mathrm{2}} \frac{\mathrm{8}}{{x}} \\ $$

Question Number 118491 Answers: 1 Comments: 0

... ⧫Advanced Calculus⧫... Evaluate:: Ω = ∫_0 ^( ∞) ((secθ)/( (√(4tan^2 θ+5))))dθ ...♠L𝛗rD ∅sE♠... ...♣GooD LucK♣

$$ \\ $$$$...\:\blacklozenge\mathrm{Advanced}\:\mathrm{Calculus}\blacklozenge... \\ $$$$ \\ $$$$\mathrm{Evaluate}:: \\ $$$$ \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{sec}\theta}{\:\sqrt{\mathrm{4tan}^{\mathrm{2}} \theta+\mathrm{5}}}\mathrm{d}\theta \\ $$$$ \\ $$$$...\spadesuit\boldsymbol{\mathrm{L}\phi\mathrm{rD}}\:\boldsymbol{\varnothing\mathrm{sE}}\spadesuit... \\ $$$$ \\ $$$$...\clubsuit\boldsymbol{\mathrm{GooD}}\:\boldsymbol{\mathrm{LucK}}\clubsuit \\ $$

Question Number 118489 Answers: 2 Comments: 0

(1) solve the equation ((x−49)/(50)) + ((x−50)/(49)) = ((49)/(x−50)) + ((50)/(x−49)) (2) How many numbers from 12 to 12345 inclusive have digits which are consecutive an in increasing order, reading from left to right ?

$$\left(\mathrm{1}\right)\:{solve}\:{the}\:{equation}\:\frac{{x}−\mathrm{49}}{\mathrm{50}}\:+\:\frac{{x}−\mathrm{50}}{\mathrm{49}}\:=\:\frac{\mathrm{49}}{{x}−\mathrm{50}}\:+\:\frac{\mathrm{50}}{{x}−\mathrm{49}} \\ $$$$\left(\mathrm{2}\right)\:{How}\:{many}\:{numbers}\:{from}\:\mathrm{12}\:{to}\:\mathrm{12345}\: \\ $$$${inclusive}\:{have}\:{digits}\:{which}\:{are}\: \\ $$$${consecutive}\:{an}\:{in}\:{increasing}\:{order}, \\ $$$${reading}\:{from}\:{left}\:{to}\:{right}\:?\: \\ $$

Question Number 118488 Answers: 1 Comments: 0

Conjecture a formula for the infinite sum of the series. (1/3)+(1/(15))+(1/(35))+ ∙ ∙ ∙ (1/((2n−1)(2n+1))) And prove the formula by Induction.

Question Number 118482 Answers: 1 Comments: 0

If the tangents at the end of a focal chord of parabola meet the tangent at the vertex in C,D.prove that CD substends a right angle at the focus

$$\mathrm{If}\:\mathrm{the}\:\mathrm{tangents}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{end}\:\mathrm{of}\:\mathrm{a}\:\mathrm{focal}\:\:\mathrm{chord}\:\mathrm{of} \\ $$$$\mathrm{parabola}\:\mathrm{meet}\:\mathrm{the} \\ $$$$\mathrm{tangent}\:\mathrm{at}\:\mathrm{the}\:\:\mathrm{vertex} \\ $$$$\mathrm{in}\:\mathrm{C},\mathrm{D}.\mathrm{prove}\:\mathrm{that}\:\mathrm{CD} \\ $$$$\mathrm{substends}\:\mathrm{a}\:\mathrm{right}\:\mathrm{angle} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{focus} \\ $$

Question Number 118478 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−2x) ln(1+3x)dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{2x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{3x}\right)\mathrm{dx} \\ $$

Question Number 118476 Answers: 0 Comments: 0

find ∫_(−∞) ^∞ ((arctan(1+2x))/(x^2 +1))dx

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

Question Number 118475 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(2+x^2 ))/(x^2 +9))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{2}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{9}}\mathrm{dx} \\ $$

Question Number 118466 Answers: 2 Comments: 0

solve ∫_0 ^1 ((tan^(−1) x)/(√(1−x^2 )))dx

$${solve} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{tan}^{−\mathrm{1}} {x}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 118455 Answers: 6 Comments: 1

Question Number 118452 Answers: 2 Comments: 3

Question: 2^x +2^(2x+1) +1=y^2 solve this equation if x,y𝛆Z

$$\boldsymbol{{Question}}: \\ $$$$\mathrm{2}^{\boldsymbol{{x}}} +\mathrm{2}^{\mathrm{2}\boldsymbol{{x}}+\mathrm{1}} +\mathrm{1}=\boldsymbol{{y}}^{\mathrm{2}} \:\:\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\:\boldsymbol{{equation}}\:\boldsymbol{{if}} \\ $$$$\boldsymbol{{x}},\boldsymbol{{y}\epsilon}\mathbb{Z} \\ $$

Question Number 118448 Answers: 3 Comments: 0

Question Number 118442 Answers: 1 Comments: 0

sin x.y′′ +2cos x. y′−y sin x = e^x

$$\:\:\mathrm{sin}\:{x}.{y}''\:+\mathrm{2cos}\:{x}.\:{y}'−{y}\:\mathrm{sin}\:{x}\:=\:{e}^{{x}} \\ $$$$ \\ $$

Question Number 118485 Answers: 2 Comments: 0

If partial fraction ((10x^2 +px+18)/(2x^3 +5x^2 +x−2)) can be written as (q/(2x−1)) + (4/(x+2)) + (r/(x+1)). Then find the value of p−q+2r .

$${If}\:{partial}\:{fraction}\: \\ $$$$\frac{\mathrm{10}{x}^{\mathrm{2}} +{px}+\mathrm{18}}{\mathrm{2}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} +{x}−\mathrm{2}}\:{can}\:{be}\:{written} \\ $$$${as}\:\frac{{q}}{\mathrm{2}{x}−\mathrm{1}}\:+\:\frac{\mathrm{4}}{{x}+\mathrm{2}}\:+\:\frac{{r}}{{x}+\mathrm{1}}.\:{Then}\:{find}\:{the} \\ $$$${value}\:{of}\:{p}−{q}+\mathrm{2}{r}\:. \\ $$

Question Number 118438 Answers: 0 Comments: 0

... nice calculus... evaluate :: lim_(s→0) ((ζ( 1+s )+ζ(1−s))/2) =^? γ γ: euler−mascheroni constant m.n.1970.

$$\:\:\:\:\:\:...\:\:{nice}\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${lim}_{{s}\rightarrow\mathrm{0}} \frac{\zeta\left(\:\mathrm{1}+{s}\:\right)+\zeta\left(\mathrm{1}−{s}\right)}{\mathrm{2}}\:\overset{?} {=}\gamma \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\gamma:\:{euler}−{mascheroni}\:{constant} \\ $$$$\:\:\:{m}.{n}.\mathrm{1970}. \\ $$$$ \\ $$

Question Number 118436 Answers: 4 Comments: 0

∫ cos^4 (x) cos^4 (2x) dx

$$\:\:\int\:\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{2}{x}\right)\:{dx}\: \\ $$

Question Number 118435 Answers: 2 Comments: 0

If 4 (x^9 )^(1/(4 )) −9 (x^9 )^(1/(8 )) + 4 = 0 , then (x^9 )^(1/(4 )) + (x^(−9) )^(1/(4 )) =?

$${If}\:\mathrm{4}\:\sqrt[{\mathrm{4}\:}]{{x}^{\mathrm{9}} }\:−\mathrm{9}\:\sqrt[{\mathrm{8}\:}]{{x}^{\mathrm{9}} }\:+\:\mathrm{4}\:=\:\mathrm{0}\:,\:{then}\: \\ $$$$\:\sqrt[{\mathrm{4}\:}]{{x}^{\mathrm{9}} }\:+\:\sqrt[{\mathrm{4}\:}]{{x}^{−\mathrm{9}} }\:=? \\ $$

Question Number 118427 Answers: 7 Comments: 0

(1) lim_(x→0) ((cos x))^(1/(x^2 )) (2) lim_(x→1) ((1+cos πx)/(x^2 −2x+1))

$$\:\:\:\left(\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{{x}^{\mathrm{2}} \:}]{\mathrm{cos}\:{x}}\: \\ $$$$\:\:\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{1}+\mathrm{cos}\:\pi{x}}{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}} \\ $$

Question Number 118419 Answers: 4 Comments: 1

∫ ((2sin 2x)/(4cos x+sin 2x)) dx

$$\:\:\:\int\:\frac{\mathrm{2sin}\:\mathrm{2}{x}}{\mathrm{4cos}\:{x}+\mathrm{sin}\:\mathrm{2}{x}}\:{dx}\: \\ $$

Question Number 118411 Answers: 1 Comments: 0

Write the vector v=(1,−2,3) as a linear combination of vectors u_1 =(1,1,1) ,u_2 =(1,2,3) and u_3 =(2,−1,1)

$${Write}\:{the}\:{vector}\:{v}=\left(\mathrm{1},−\mathrm{2},\mathrm{3}\right)\:{as}\:{a} \\ $$$${linear}\:{combination}\:{of}\:{vectors} \\ $$$${u}_{\mathrm{1}} =\left(\mathrm{1},\mathrm{1},\mathrm{1}\right)\:,{u}_{\mathrm{2}} =\left(\mathrm{1},\mathrm{2},\mathrm{3}\right)\:{and}\:{u}_{\mathrm{3}} =\left(\mathrm{2},−\mathrm{1},\mathrm{1}\right) \\ $$

Question Number 118395 Answers: 2 Comments: 5

∫_a ^b ((f(x))/(f(x)+f(a+b−x)))dx

$$\int_{{a}} ^{{b}} \frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx} \\ $$

Question Number 118391 Answers: 2 Comments: 0

show that if ^ a^2 +b^2 can be divised by 7, a+b can also be divised by 7.

$${show}\:{that}\:{if}\:\:^{} {a}^{\mathrm{2}} +{b}^{\mathrm{2}} \:{can}\:{be}\:{divised} \\ $$$${by}\:\mathrm{7},\:{a}+{b}\:{can}\:{also}\:{be}\:{divised}\:{by}\:\mathrm{7}. \\ $$

Question Number 118381 Answers: 3 Comments: 1

How do you express this question in partial fraction ((5x^2 +x+6)/((3−2x)(x^2 +4))) hence obtain the expansion is ascending powers of x up to and including the term x^2

$${How}\:{do}\:{you}\:{express}\:{this}\:{question} \\ $$$${in}\:{partial}\:{fraction}\:\frac{\mathrm{5}{x}^{\mathrm{2}} +{x}+\mathrm{6}}{\left(\mathrm{3}−\mathrm{2}{x}\right)\left({x}^{\mathrm{2}} +\mathrm{4}\right)}\: \\ $$$${hence}\:{obtain}\:{the}\:{expansion} \\ $$$${is}\:{ascending}\:{powers}\:{of}\:{x}\:{up} \\ $$$${to}\:{and}\:{including}\:{the}\:{term}\:{x}^{\mathrm{2}} \\ $$

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