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Question Number 44604 Answers: 1 Comments: 1

∫[((log x − 1)/(1+(log x)^2 ))]^2 dx = (x/((log x)^2 +1))+C

$$\int\left[\frac{\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\:\:−\:\:\mathrm{1}}{\mathrm{1}+\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} }\right]^{\mathrm{2}} \boldsymbol{\mathrm{dx}}\:\:=\:\:\frac{\boldsymbol{\mathrm{x}}}{\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} +\mathrm{1}}+\boldsymbol{\mathrm{C}} \\ $$

Question Number 44602 Answers: 0 Comments: 0

∫(e^x /(1+x^2 )) dx = ?

$$\int\frac{\boldsymbol{{e}}^{\boldsymbol{\mathrm{x}}} }{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}}\:=\:\:? \\ $$

Question Number 44587 Answers: 1 Comments: 2

calculate ∫_0 ^∞ (dt/(1+t^(2018) ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2018}} } \\ $$

Question Number 44584 Answers: 1 Comments: 5

Prove that if a, b, c ∈ Z and a^2 + b^2 = c^2 , then 3 ∣ ab

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\:{a},\:{b},\:{c}\:\in\:\mathbb{Z}\:\:\mathrm{and}\:\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:{c}^{\mathrm{2}} ,\:\mathrm{then} \\ $$$$\mathrm{3}\:\mid\:{ab} \\ $$

Question Number 44674 Answers: 0 Comments: 1

solving some integrals we might meet some of the following functions which cannot be solved with elementar knowledge but tables should exist somewhere in the depth of the www... these links might be interesting exponential integral ∫(e^(−x) /x)dx en.wikipedia.org/wiki/Exponential_integral logarithmic integral ∫(dx/(ln x)) en.wikipedia.org/wiki/Logarithmic_integral_function also see en.wikipedia.org/wiki/Polylogarithm trigonometric integrals i.e. ∫((sin x)/x)dx en.wikipedia.org/wiki/Trigonometric_integral

$$\mathrm{solving}\:\mathrm{some}\:\mathrm{integrals}\:\mathrm{we}\:\mathrm{might}\:\mathrm{meet}\:\mathrm{some} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{functions}\:\mathrm{which}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{solved}\:\mathrm{with}\:\mathrm{elementar}\:\mathrm{knowledge}\:\mathrm{but}\:\mathrm{tables} \\ $$$$\mathrm{should}\:\mathrm{exist}\:\mathrm{somewhere}\:\mathrm{in}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{www}... \\ $$$$\mathrm{these}\:\mathrm{links}\:\mathrm{might}\:\mathrm{be}\:\mathrm{interesting} \\ $$$$ \\ $$$$\mathrm{exponential}\:\mathrm{integral} \\ $$$$\int\frac{\mathrm{e}^{−{x}} }{{x}}{dx} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Exponential\_integral} \\ $$$$ \\ $$$$\mathrm{logarithmic}\:\mathrm{integral} \\ $$$$\int\frac{{dx}}{\mathrm{ln}\:{x}} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Logarithmic\_integral\_function} \\ $$$$\mathrm{also}\:\mathrm{see} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Polylogarithm} \\ $$$$ \\ $$$$\mathrm{trigonometric}\:\mathrm{integrals} \\ $$$$\mathrm{i}.\mathrm{e}.\:\int\frac{\mathrm{sin}\:{x}}{{x}}{dx} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Trigonometric\_integral} \\ $$

Question Number 44570 Answers: 1 Comments: 0

Question Number 44573 Answers: 1 Comments: 1

Question Number 44575 Answers: 1 Comments: 3

Question Number 44557 Answers: 1 Comments: 2

Question Number 44548 Answers: 0 Comments: 0

(a) The area of a sector of a circle of radius 12cm is 132cm^2 . If the sector is folded such that its straight edges coincide to form a cone. Find the radius of the base of the cone [ Take π = ((22)/7) ] . (b) A circle of centre O has radius 5cm. A chord PQ of the circle is 6cm long. caclculate: (i) The distance of the chord from the centre O (ii) The angle POQ

$$\left(\mathrm{a}\right) \\ $$$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{sector}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:\:\mathrm{12cm}\:\mathrm{is}\:\:\mathrm{132cm}^{\mathrm{2}} \:.\:\:\mathrm{If}\:\mathrm{the}\:\mathrm{sector} \\ $$$$\mathrm{is}\:\mathrm{folded}\:\mathrm{such}\:\mathrm{that}\:\mathrm{its}\:\mathrm{straight}\:\mathrm{edges}\:\mathrm{coincide}\:\mathrm{to}\:\mathrm{form}\:\mathrm{a}\:\mathrm{cone}.\:\mathrm{Find}\:\mathrm{the}\: \\ $$$$\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{base}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\:\:\:\left[\:\:\mathrm{Take}\:\:\:\:\pi\:\:=\:\:\frac{\mathrm{22}}{\mathrm{7}}\:\right]\:. \\ $$$$ \\ $$$$\left(\mathrm{b}\right)\:\:\: \\ $$$$\mathrm{A}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{centre}\:\mathrm{O}\:\mathrm{has}\:\mathrm{radius}\:\mathrm{5cm}.\:\:\mathrm{A}\:\mathrm{chord}\:\mathrm{PQ}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{6cm}\:\mathrm{long}. \\ $$$$\mathrm{caclculate}: \\ $$$$\:\:\:\:\left(\mathrm{i}\right)\:\:\:\mathrm{The}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{the}\:\mathrm{chord}\:\mathrm{from}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{O} \\ $$$$\:\:\:\left(\mathrm{ii}\right)\:\:\mathrm{The}\:\mathrm{angle}\:\mathrm{POQ} \\ $$

Question Number 44546 Answers: 1 Comments: 0

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Question Number 44543 Answers: 1 Comments: 3

If y =f(x) = ax^2 +bx+c and at some x, say x= p ∫_0 ^( p) ydx = y(p)= y ′(p) = y ′′(p)= p , then find p .

$${If}\:\:{y}\:={f}\left({x}\right)\:=\:{ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${and}\:\:{at}\:{some}\:{x},\:{say}\:\:{x}=\:{p} \\ $$$$\int_{\mathrm{0}} ^{\:\:{p}} {ydx}\:=\:{y}\left({p}\right)=\:{y}\:'\left({p}\right)\:=\:{y}\:''\left({p}\right)=\:{p}\:, \\ $$$${then}\:{find}\:\boldsymbol{{p}}\:. \\ $$

Question Number 44541 Answers: 1 Comments: 1

Find the remainder when the polynomial p(y)=y^4 −3y^2 +2y+1 is divided by y−1.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{the} \\ $$$$\mathrm{polynomial}\:{p}\left({y}\right)={y}^{\mathrm{4}} −\mathrm{3}{y}^{\mathrm{2}} +\mathrm{2}{y}+\mathrm{1}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:{y}−\mathrm{1}. \\ $$

Question Number 44537 Answers: 1 Comments: 0

Question Number 44535 Answers: 0 Comments: 1

Question Number 44527 Answers: 0 Comments: 2

Question Number 44526 Answers: 1 Comments: 1

Find moment of inertia of the area bounded by the curve r^2 =a^2 cos2θ about its axis

$$\mathrm{Find}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{inertia}\:\mathrm{of}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{r}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} \mathrm{cos2}\theta \\ $$$$\mathrm{about}\:\mathrm{its}\:\mathrm{axis} \\ $$

Question Number 44515 Answers: 1 Comments: 0

let g(x) =∫_0 ^∞ ((t ln(t)dt)/((1+xt)^3 )) with x>0 1) give a explicit form of g(x) 2) calculate ∫_0 ^∞ ((t ln(t))/((1+t)^3 ))dt 3) calculate ∫_0 ^∞ ((tln(t))/((1+2t)^3 )) dt 4) calculate A(θ) =∫_0 ^∞ ((t ln(t))/((1+t sinθ)^3 ))dt with 0<θ<(π/2)

$${let}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}\:{ln}\left({t}\right){dt}}{\left(\mathrm{1}+{xt}\right)^{\mathrm{3}} }\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{explicit}\:{form}\:{of}\:{g}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}\:{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)^{\mathrm{3}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{tln}\left({t}\right)}{\left(\mathrm{1}+\mathrm{2}{t}\right)^{\mathrm{3}} }\:{dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}\:{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\:{sin}\theta\right)^{\mathrm{3}} }{dt}\:\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 44512 Answers: 1 Comments: 1

prove that:−∫2^(ln x) dx = ((x.2^(ln x) )/(ln(xe))) +C

$$\boldsymbol{{prove}}\:\boldsymbol{{that}}:−\int\mathrm{2}^{\boldsymbol{\mathrm{ln}}\:\boldsymbol{\mathrm{x}}} \:\boldsymbol{\mathrm{dx}}\:=\:\frac{\boldsymbol{\mathrm{x}}.\mathrm{2}^{\boldsymbol{\mathrm{ln}}\:\boldsymbol{\mathrm{x}}} }{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{xe}}\right)}\:+\boldsymbol{\mathrm{C}} \\ $$$$ \\ $$

Question Number 44509 Answers: 1 Comments: 1

∫(√(tan x)) dx=?

$$\int\sqrt{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\mathrm{x}}}\:\:\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 44508 Answers: 1 Comments: 1

∫(√(sin x ))dx=?

$$\int\sqrt{\boldsymbol{\mathrm{sin}}\:\boldsymbol{\mathrm{x}}\:}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 44502 Answers: 1 Comments: 0

If a>b,and c>d,prove that a−c may be greater than, equal to or less than b−d.