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Question Number 38568 Answers: 0 Comments: 18

Find the area common to min{[x], [y] } =2 and max{[x], [y] } =4 . [x] denotes the greatest integer less than or equal to x.

$${Find}\:{the}\:{area}\:{common}\:{to} \\ $$$${min}\left\{\left[{x}\right],\:\left[{y}\right]\:\right\}\:=\mathrm{2}\:\:\:{and} \\ $$$${max}\left\{\left[{x}\right],\:\left[{y}\right]\:\right\}\:=\mathrm{4}\:. \\ $$$$\left[{x}\right]\:{denotes}\:{the}\:{greatest}\:{integer} \\ $$$${less}\:{than}\:{or}\:{equal}\:{to}\:{x}. \\ $$

Question Number 38562 Answers: 1 Comments: 0

((√2) +i)(1−(√(2i)) )

$$\left(\sqrt{\mathrm{2}}\:+{i}\right)\left(\mathrm{1}−\sqrt{\mathrm{2}{i}}\:\right) \\ $$

Question Number 38559 Answers: 1 Comments: 0

in a geometric series, the first term =a, common ratio=r. If S_n denotes the sum of the n terms and U_n =Σ_(n=1) ^n S_(n,) then rS_n +(1−r)U_(n ) equals to (a) 0 (b) n (c) na (d)nar

$${in}\:{a}\:{geometric}\:{series},\:{the}\:{first}\:{term} \\ $$$$={a},\:{common}\:{ratio}={r}.\:{If}\:{S}_{{n}} \:{denotes} \\ $$$${the}\:{sum}\:{of}\:{the}\:{n}\:{terms}\:{and}\:{U}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}{S}_{{n},} \\ $$$${then}\:{rS}_{{n}} +\left(\mathrm{1}−{r}\right){U}_{{n}\:\:} {equals}\:{to} \\ $$$$\left({a}\right)\:\:\mathrm{0}\:\:\:\:\:\:\left({b}\right)\:\:{n}\:\:\:\:\:\left({c}\right)\:\:\:\:{na}\:\:\:\:\left({d}\right){nar} \\ $$

Question Number 38557 Answers: 1 Comments: 0

prove that ((2 cos 2^n θ + 1)/(2 cos θ + 1)) = (2 cos θ − 1)(2 cos 2θ − 1)(2 cos 2^2 θ− 1) ...(2 cos 2^(n − 1) θ − 1)

Question Number 38536 Answers: 2 Comments: 0

∫_0 ^π (dx/(√(3−cos x)))=

$$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\sqrt{\mathrm{3}−\mathrm{cos}\:{x}}}= \\ $$

Question Number 38535 Answers: 0 Comments: 0

Given the function f(x) where f(x)= { ((∫x^2 + 1 ,for {x:x D(f) 2)),((∫x^3 − 1,for y = f′(x))) :} a) Evaluate f(2) if f(a)= 2 + a^(n−1) find the value of a hence the domain of f(x).

$${Given}\:{the}\:{function} \\ $$$${f}\left({x}\right)\:{where}\: \\ $$$$ \\ $$$${f}\left({x}\right)=\:\begin{cases}{\int{x}^{\mathrm{2}} \:+\:\mathrm{1}\:,{for}\:\left\{{x}:{x}\:{D}\left({f}\right)\:\mathrm{2}\right.}\\{\int{x}^{\mathrm{3}} \:−\:\mathrm{1},{for}\:{y}\:=\:{f}'\left({x}\right)}\end{cases} \\ $$$$\left.{a}\right)\:{Evaluate}\:{f}\left(\mathrm{2}\right) \\ $$$${if}\:{f}\left({a}\right)=\:\mathrm{2}\:+\:{a}^{{n}−\mathrm{1}} \\ $$$${find}\:{the}\:{value}\:{of}\:{a} \\ $$$${hence}\:{the}\:{domain}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 38534 Answers: 0 Comments: 0

∫∫_R (2x + 3y)^2 dA=??

$$\int\underset{{R}} {\int}\left(\mathrm{2}{x}\:+\:\mathrm{3}{y}\right)^{\mathrm{2}} \:{dA}=?? \\ $$

Question Number 38533 Answers: 2 Comments: 2

Question Number 38521 Answers: 0 Comments: 1

letf(x) = ((2x+1)/((x−2)(x^2 +x+1))) 1) calculate f^((n)) (x) 2) find f^((n)) (0) 3) developp f at integr serie. (

$${letf}\left({x}\right)\:=\:\frac{\mathrm{2}{x}+\mathrm{1}}{\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$$$\left(\right. \\ $$

Question Number 38520 Answers: 0 Comments: 1

find the value of Σ_(n=2) ^∞ ((3n^2 +1)/((n−1)^3 (n+1)^3 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\frac{\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 38518 Answers: 0 Comments: 1

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

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

Question Number 38517 Answers: 2 Comments: 0

simlify A= (1/((2−(√5))^4 )) + (1/((2+(√5))^4 )) B = (1/((3−(√2))^6 )) +(1/((3+(√2))^6 ))

$${simlify} \\ $$$${A}=\:\frac{\mathrm{1}}{\left(\mathrm{2}−\sqrt{\mathrm{5}}\right)^{\mathrm{4}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)^{\mathrm{4}} } \\ $$$${B}\:=\:\frac{\mathrm{1}}{\left(\mathrm{3}−\sqrt{\mathrm{2}}\right)^{\mathrm{6}} }\:+\frac{\mathrm{1}}{\left(\mathrm{3}+\sqrt{\mathrm{2}}\right)^{\mathrm{6}} } \\ $$

Question Number 38516 Answers: 3 Comments: 1

∫_0 ^(π/2) ∣sin x − cos x∣dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mid\mathrm{sin}\:{x}\:−\:\mathrm{cos}\:{x}\mid\mathrm{d}{x} \\ $$

Question Number 38515 Answers: 1 Comments: 0

Question ; x^3 + x^3 = A) x^9 B) x^6 C) x^3 D) 1 Give a reason for your answer.

$$\:{Question}\:; \\ $$$${x}^{\mathrm{3}} \:+\:{x}^{\mathrm{3}} \:=\: \\ $$$$\left.{A}\right)\:{x}^{\mathrm{9}} \\ $$$$\left.{B}\right)\:{x}^{\mathrm{6}} \\ $$$$\left.{C}\right)\:{x}^{\mathrm{3}} \\ $$$$\left.{D}\right)\:\mathrm{1} \\ $$$${Give}\:{a}\:{reason}\:{for}\:{your}\:{answer}. \\ $$

Question Number 38492 Answers: 0 Comments: 1

Question Number 38495 Answers: 4 Comments: 0

prove that tan 3a tan 2a tan a = tan 3a − tan 2a − tan a

$${prove}\:{that} \\ $$$$\boldsymbol{\mathrm{tan}}\:\mathrm{3}\boldsymbol{{a}}\:\boldsymbol{\mathrm{tan}}\:\mathrm{2}\boldsymbol{{a}}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{{a}}\:=\:\:\boldsymbol{\mathrm{tan}}\:\mathrm{3}\boldsymbol{{a}}\:−\:\boldsymbol{\mathrm{tan}}\:\mathrm{2}\boldsymbol{{a}}\:−\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{{a}} \\ $$

Question Number 38488 Answers: 2 Comments: 0

find the value of x if 3^x = 9x

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{if}\: \\ $$$$\mathrm{3}^{{x}} \:=\:\mathrm{9}{x} \\ $$

Question Number 38478 Answers: 0 Comments: 1

Question Number 38477 Answers: 1 Comments: 0

Question Number 38475 Answers: 1 Comments: 0

A committee of 2 girls and 3boys is to be form from 6girls and 8boys how many different committee can be formed ?

$${A}\:{committee}\:{of}\:\mathrm{2}\:{girls}\:{and}\:\mathrm{3}{boys} \\ $$$${is}\:{to}\:{be}\:{form}\:{from}\:\mathrm{6}{girls}\:{and}\:\mathrm{8}{boys} \\ $$$${how}\:{many}\:{different}\:{committee}\:{can} \\ $$$${be}\:{formed} \\ $$$$? \\ $$

Question Number 38470 Answers: 0 Comments: 4

calculate f(t)=∫_0 ^∞ ((cos(tx))/((1+tx^2 )^2 )) dx with t≥0 2) find the values of ∫_0 ^∞ ((cos(2x))/((1+2x^2 )^2 ))dx and ∫_0 ^∞ ((cosx)/((2+x^2 )^2 ))dx

$${calculate}\:{f}\left({t}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({tx}\right)}{\left(\mathrm{1}+{tx}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:{with}\:{t}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cosx}}{\left(\mathrm{2}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 38469 Answers: 0 Comments: 1

calculate f(a) = ∫_(−∞) ^(+∞) ((sin(ax))/(x^2 +x+1))dx 2) find the value of ∫_(−∞) ^(+∞) ((sin(3x))/(x^2 +x+1))dx

$${calculate}\:\:{f}\left({a}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({ax}\right)}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left(\mathrm{3}{x}\right)}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}{dx} \\ $$

Question Number 38468 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((sin(2x)sh(3x))/(4+x^2 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left(\mathrm{2}{x}\right){sh}\left(\mathrm{3}{x}\right)}{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 38467 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((cos(ax)ch(bx))/(x^2 +1))dx .

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{cos}\left({ax}\right){ch}\left({bx}\right)}{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx}\:. \\ $$

Question Number 38466 Answers: 0 Comments: 1

let a from R find F_a (t)= ∫_(−∞) ^(+∞) ((cos(tx))/(a^2 +x^2 ))dx 2) calculate F_2 (3) and F_3 (2)

$${let}\:{a}\:{from}\:{R}\:\:{find}\:{F}_{{a}} \left({t}\right)=\:\int_{−\infty} ^{+\infty} \:\:\frac{{cos}\left({tx}\right)}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{F}_{\mathrm{2}} \left(\mathrm{3}\right)\:\:{and}\:{F}_{\mathrm{3}} \left(\mathrm{2}\right) \\ $$

Question Number 38465 Answers: 0 Comments: 2

find f(x)= ∫_0 ^1 ln(1+xt^3 )dt with ∣x∣<1 . 2) calculate ∫_0 ^1 ln(1+4t^3 )dt and ∫_0 ^1 ln(2+t^3 )dt.

$${find}\:\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{3}} \right){dt}\:{with}\:\mid{x}\mid<\mathrm{1}\:. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+\mathrm{4}{t}^{\mathrm{3}} \right){dt}\:\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{2}+{t}^{\mathrm{3}} \right){dt}. \\ $$

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