Question and Answers Forum

AllQuestion and Answers: Page 1708

Question Number 32775 Answers: 1 Comments: 0

Find the area bounded by the curve y=3x^2 +2x−3,the x axis and the line x=5 and x=2

$${Find}\:{the}\:{area}\:{bounded}\:{by}\:{the} \\ $$$${curve}\:{y}=\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{3},{the}\:{x}\:{axis} \\ $$$${and}\:{the}\:{line}\:{x}=\mathrm{5}\:{and}\:{x}=\mathrm{2} \\ $$

Question Number 32768 Answers: 1 Comments: 0

Prove that a^2 +b^2 +c^2 ≥ab+bc+ca ∀ a,b,c∈R

$$\mathrm{Prove}\:\mathrm{that}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \geqslant{ab}+{bc}+{ca} \\ $$$$\forall\:{a},{b},{c}\in\mathbb{R} \\ $$

Question Number 32767 Answers: 0 Comments: 2

For a,b,c≥0 if a+b+c=n, determine minimum and maximum values of a^2 +b^2 +c^2 −ab−bc−ca.

$$\mathrm{For}\:{a},{b},{c}\geqslant\mathrm{0}\:\mathrm{if}\:{a}+{b}+{c}={n},\:\mathrm{determine} \\ $$$$\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{values}\:\mathrm{of} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}. \\ $$

Question Number 32765 Answers: 0 Comments: 0

Question Number 32763 Answers: 1 Comments: 1

Question Number 32760 Answers: 1 Comments: 0

if y=3x^(4 ) .find the approximate percentage increase in y when x increase by 2(1/2)%.

$$\mathrm{if}\:{y}=\mathrm{3}{x}^{\mathrm{4}\:} .{find}\:{the}\:{approximate}\:{percentage} \\ $$$${increase}\:{in}\:{y}\:{when}\:{x}\:{increase}\:{by}\:\:\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}}\%. \\ $$

Question Number 32757 Answers: 1 Comments: 5

A mass oscillating on a spring has amplitude of 1.2m and a period of 2s.Deduce the equation for the displacement x if the timing starts at the instant when the masd has its maximum displacement. b)calculate the time interval from t=0 before the displacement is 0.08m

$${A}\:{mass}\:{oscillating}\:{on}\:{a}\:{spring} \\ $$$${has}\:{amplitude}\:{of}\:\mathrm{1}.\mathrm{2}{m}\:{and}\:{a}\:{period} \\ $$$${of}\:\mathrm{2}{s}.{Deduce}\:{the}\:{equation}\:{for} \\ $$$${the}\:{displacement}\:{x}\:{if}\:{the}\:{timing} \\ $$$${starts}\:{at}\:{the}\:{instant}\:{when}\:{the}\:{masd} \\ $$$${has}\:{its}\:{maximum}\:{displacement}. \\ $$$$\left.{b}\right){calculate}\:{the}\:{time}\:{interval}\:{from} \\ $$$${t}=\mathrm{0}\:{before}\:{the}\:{displacement}\:{is} \\ $$$$\mathrm{0}.\mathrm{08}{m} \\ $$$$ \\ $$

Question Number 32756 Answers: 1 Comments: 0

Please help Find the area bounded by y(x+2)=x^4 ,x=0,y=0,and x=3

$${Please}\:{help} \\ $$$$ \\ $$$${Find}\:{the}\:{area}\:{bounded}\:{by} \\ $$$${y}\left({x}+\mathrm{2}\right)={x}^{\mathrm{4}} ,{x}=\mathrm{0},{y}=\mathrm{0},{and}\:{x}=\mathrm{3} \\ $$

Question Number 32755 Answers: 0 Comments: 5

Σ_(n=1) ^∞ (1/n^2 ) = ....???

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:=\:....??? \\ $$

Question Number 32745 Answers: 1 Comments: 0

The first, second and middle terms of an AP are a, b, c respectively. Their sum is

$$\mathrm{The}\:\mathrm{first},\:\mathrm{second}\:\mathrm{and}\:\mathrm{middle}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{AP}\:\mathrm{are}\:{a},\:{b},\:{c}\:\mathrm{respectively}.\:\mathrm{Their}\:\mathrm{sum}\:\mathrm{is} \\ $$

Question Number 32743 Answers: 2 Comments: 1

f (f (n)) = 2n f (n) = ?

$${f}\:\left({f}\:\left({n}\right)\right)\:\:=\:\:\mathrm{2}{n} \\ $$$${f}\:\left({n}\right)\:\:=\:\:? \\ $$

Question Number 32741 Answers: 0 Comments: 0

find ∫_0 ^1 ((ln(t^2 +2t cosx +1))/t)dt .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left({t}^{\mathrm{2}} \:+\mathrm{2}{t}\:{cosx}\:+\mathrm{1}\right)}{{t}}{dt}\:. \\ $$

Question Number 32740 Answers: 0 Comments: 2

find∫_0 ^∞ ((ln(x^2 +t^2 ))/(1+t^2 ))dt

$${find}\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 32739 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ (e^(−t) /(1+xt))dt calculate f^((n)) (0).

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{t}} }{\mathrm{1}+{xt}}{dt} \\ $$$${calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right). \\ $$

Question Number 32737 Answers: 1 Comments: 0

let give 0≤x≤1 calculate ∫_0 ^∞ ((arctan((x/t)))/(1+t^2 )) dt

$${let}\:{give}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \frac{{arctan}\left(\frac{{x}}{{t}}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$

Question Number 32736 Answers: 0 Comments: 0

let o≤x≤1 find ∫_0 ^x ((lnt)/(t^2 −1))dt

$${let}\:{o}\leqslant{x}\leqslant\mathrm{1}\:\:{find}\:\int_{\mathrm{0}} ^{{x}} \:\frac{{lnt}}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\: \\ $$

Question Number 32735 Answers: 0 Comments: 0

let give A_n =∫_0 ^1 (dt/(1+t^n )) 1) find l=lim_(n→∞) A_n 2)give a equivalent of A_n −l 3) find a equivalent of A_n

$${let}\:{give}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{1}+{t}^{{n}} } \\ $$$$\left.\mathrm{1}\right)\:{find}\:{l}={lim}_{{n}\rightarrow\infty} \:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right){give}\:{a}\:{equivalent}\:{of}\:{A}_{{n}} −{l} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{A}_{{n}} \\ $$

Question Number 32734 Answers: 1 Comments: 3

1) a≥0 calculate ∫_0 ^a ((n^2 −x^2 )/((n^2 +x^2 )^2 ))dx with n integr 2) find ∫_0 ^∞ ((n^2 −x^2 )/((n^2 +x^2 )^2 ))dx 3)calculate Σ_(n=1) ^∞ ∫_0 ^∞ ((n^2 −x^2 )/((n^2 +x^2 )^2 )) dx .

$$\left.\mathrm{1}\right)\:{a}\geqslant\mathrm{0}\:\:{calculate}\:\int_{\mathrm{0}} ^{{a}} \:\frac{{n}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }{\left({n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:{with} \\ $$$${n}\:{integr} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{n}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }{\left({n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right){calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }{\left({n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 32733 Answers: 0 Comments: 0

prove that Σ_(n=0) ^∞ (1/((n!)^2 )) =(1/(2π)) ∫_0 ^(2π) e^(2cosx) dx .

$${prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left({n}!\right)^{\mathrm{2}} }\:=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:{e}^{\mathrm{2}{cosx}} {dx}\:. \\ $$

Question Number 32732 Answers: 0 Comments: 0

give ∫_0 ^1 (((lnx)^p )/(1−x)) dx at form of seriewith p≥2 .

$${give}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\left({lnx}\right)^{{p}} }{\mathrm{1}−{x}}\:{dx}\:{at}\:{form}\:{of}\:{seriewith} \\ $$$${p}\geqslant\mathrm{2}\:. \\ $$

Question Number 32731 Answers: 0 Comments: 0

1) prove that ∫_0 ^1 ((arctant)/t)dt=−∫_0 ^1 ((lnt)/(1+t^2 ))dt 2) find ∫_0 ^1 ((arctant)/t)dt at form of serie

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctant}}{{t}}{dt}=−\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{lnt}}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctant}}{{t}}{dt}\:{at}\:{form}\:{of}\:{serie} \\ $$

Question Number 32729 Answers: 0 Comments: 0

find lim_(n→∞) ∫_0 ^n (cos((x/n)))^n^2 dx.