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Question Number 83719 Answers: 2 Comments: 1

Question. ^(Show that ∫_0 ^(Π/2) ((cosx)/(3+cos^2 x))dx=(1/4)ln3)

$${Question}.\:\:\:\:\:\:\:\:\overset{{Show}\:\:{that}\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \frac{{cosx}}{\mathrm{3}+{cos}^{\mathrm{2}} {x}}{dx}=\frac{\mathrm{1}}{\mathrm{4}}{ln}\mathrm{3}} {\:} \\ $$

Question Number 83604 Answers: 0 Comments: 7

need help. When typing with microsoft word i face some difficulties like when typing lim_(x→0) f(x) it turns to lim_(x→0) f(x) and Σ_(r=0) ^n a_n turns to Σ_(r=0) ^n a_n please how do i rectify this problem? and any suggestion on a better application to type my maths papers? thanks in advance.

$$\mathrm{need}\:\mathrm{help}.\:\mathrm{When}\:\mathrm{typing}\:\mathrm{with}\:\mathrm{microsoft}\:\mathrm{word} \\ $$$$\mathrm{i}\:\mathrm{face}\:\mathrm{some}\:\mathrm{difficulties}\:\mathrm{like}\:\mathrm{when}\:\mathrm{typing}\: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)\:\mathrm{it}\:\mathrm{turns}\:\mathrm{to}\:\mathrm{lim}_{{x}\rightarrow\mathrm{0}} \:{f}\left({x}\right)\:\mathrm{and}\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{n}} \: \\ $$$$\mathrm{turns}\:\mathrm{to}\:\sum_{{r}=\mathrm{0}} ^{{n}} {a}_{{n}} \:\:\mathrm{please}\:\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{rectify}\:\mathrm{this} \\ $$$$\mathrm{problem}?\:\mathrm{and}\:\mathrm{any}\:\mathrm{suggestion}\:\mathrm{on}\:\mathrm{a}\:\mathrm{better}\:\mathrm{application} \\ $$$$\mathrm{to}\:\mathrm{type}\:\mathrm{my}\:\mathrm{maths}\:\mathrm{papers}?\:\mathrm{thanks}\:\mathrm{in}\:\mathrm{advance}. \\ $$

Question Number 83381 Answers: 0 Comments: 0

Given that the function f(x) = x^3 is differentiable in the interval (−2,2) us the mean value theorem to find the value of x for which the tangent to the curve is parrallel to the chord through the points (−2,8) and (2,8).

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:\mathrm{is}\: \\ $$$$\mathrm{differentiable}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}\:\left(−\mathrm{2},\mathrm{2}\right)\:\mathrm{us}\:\mathrm{the}\:\mathrm{mean} \\ $$$$\mathrm{value}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\: \\ $$$$\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{is}\:\mathrm{parrallel}\:\mathrm{to}\:\mathrm{the}\:\mathrm{chord}\: \\ $$$$\mathrm{through}\:\mathrm{the}\:\mathrm{points}\:\left(−\mathrm{2},\mathrm{8}\right)\:\mathrm{and}\:\left(\mathrm{2},\mathrm{8}\right). \\ $$

Question Number 83297 Answers: 1 Comments: 1

Write down a series expansion for ln [((1−2x)/((1+2x)^2 ))] in ascending powers of x up to and including the term in x^4 . if x is small that terms in x^2 and higher powers are negleted show that (((1−2x)/(1+2x)))^(1/(2x)) ≅ (1 + x)e^(−3)

$$\mathrm{Write}\:\mathrm{down}\:\mathrm{a}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\: \\ $$$$\:\mathrm{ln}\:\left[\frac{\mathrm{1}−\mathrm{2}{x}}{\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} }\right]\:\mathrm{in}\:\mathrm{ascending}\:\mathrm{powers}\:\mathrm{of}\:\mathrm{x}\: \\ $$$$\mathrm{up}\:\mathrm{to}\:\mathrm{and}\:\mathrm{including}\:\mathrm{the}\:\mathrm{term}\:\mathrm{in}\:{x}^{\mathrm{4}} .\: \\ $$$$\mathrm{if}\:\mathrm{x}\:\mathrm{is}\:\mathrm{small}\:\mathrm{that}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{higher}\:\mathrm{powers} \\ $$$$\mathrm{are}\:\mathrm{negleted}\:\mathrm{show}\:\mathrm{that}\:\:\:\left(\frac{\mathrm{1}−\mathrm{2}{x}}{\mathrm{1}+\mathrm{2}{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}{x}}} \:\cong\:\left(\mathrm{1}\:+\:{x}\right){e}^{−\mathrm{3}} \\ $$$$ \\ $$

Question Number 83296 Answers: 0 Comments: 2

Obtain a maclaurin expansion for a) e^(cos x ) b) e^(cos^2 x)

$$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{maclaurin}\:\mathrm{expansion}\:\mathrm{for}\: \\ $$$$\left.\mathrm{a}\left.\right)\:\mathrm{e}^{\mathrm{cos}\:{x}\:} \:\:\:\:\:\:\:\:\mathrm{b}\right)\:\mathrm{e}^{\mathrm{cos}\:^{\mathrm{2}} {x}} \\ $$

Question Number 83205 Answers: 0 Comments: 2

∫_0 ^(ln2) (1/(cosh(x + ln4)))dx =

$$\int_{\mathrm{0}} ^{{ln}\mathrm{2}} \frac{\mathrm{1}}{{cosh}\left({x}\:+\:{ln}\mathrm{4}\right)}{dx}\:= \\ $$

Question Number 83202 Answers: 0 Comments: 4

find the first 4 terms in the maclaurin[ series expansion for ln (1 + 3x) hence show that if x^2 and higher powers of x are negleted, then (1 + 3x)^(3/x) = e^6 (1 −9x)

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{4}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{maclaurin}\left[\right. \\ $$$$\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\mathrm{ln}\:\left(\mathrm{1}\:+\:\mathrm{3}{x}\right)\:\mathrm{hence}\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{if}\:{x}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{higher}\:\mathrm{powers}\:\mathrm{of}\:{x}\:\mathrm{are}\:\mathrm{negleted}, \\ $$$$\mathrm{then}\: \\ $$$$\:\:\:\left(\mathrm{1}\:+\:\mathrm{3}{x}\right)^{\frac{\mathrm{3}}{{x}}} \:=\:{e}^{\mathrm{6}} \left(\mathrm{1}\:−\mathrm{9}{x}\right) \\ $$

Question Number 82929 Answers: 2 Comments: 0

if 2B+A=45° show that; tan B= ((1−2tanA−tan^2 A)/(1+2tanA−tan^2 A))

$${if}\:\mathrm{2}{B}+{A}=\mathrm{45}° \\ $$$${show}\:{that}; \\ $$$${tan}\:{B}=\:\frac{\mathrm{1}−\mathrm{2}{tanA}−{tan}^{\mathrm{2}} {A}}{\mathrm{1}+\mathrm{2}{tanA}−{tan}^{\mathrm{2}} {A}} \\ $$

Question Number 82886 Answers: 0 Comments: 2

prove (tanx+cot^2 x)^2 =sex^2 x+cosec^2 x

$${prove}\: \\ $$$$\left({tanx}+{cot}^{\mathrm{2}} {x}\right)^{\mathrm{2}} ={sex}^{\mathrm{2}} {x}+{cosec}^{\mathrm{2}} {x} \\ $$

Question Number 82867 Answers: 0 Comments: 3

hello prove that ∫_0 ^(+∞) sin(x^4 )dx=sin((π/8))∫_0 ^(+∞) e^(−x^4 ) dx? verry nice day Good Bless You

$${hello}\:{prove}\:{that}\:\int_{\mathrm{0}} ^{+\infty} {sin}\left({x}^{\mathrm{4}} \right){dx}={sin}\left(\frac{\pi}{\mathrm{8}}\right)\int_{\mathrm{0}} ^{+\infty} {e}^{−{x}^{\mathrm{4}} } {dx}? \\ $$$${verry}\:{nice}\:{day}\:{Good}\:{Bless}\:{You} \\ $$

Question Number 82729 Answers: 1 Comments: 2