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Question Number 66830 Answers: 0 Comments: 4

evaluate. ∫_1 ^( ∞) (1/x^(2 ) ) dx. can i assume lim_(t→0) ∫_1 ^( t) (1/x^(2 ) ) dx ????

$${evaluate}. \\ $$$$\:\int_{\mathrm{1}} ^{\:\infty} \:\frac{\mathrm{1}}{{x}^{\mathrm{2}\:} }\:{dx}. \\ $$$$ \\ $$$${can}\:{i}\:{assume}\:\underset{{t}\rightarrow\mathrm{0}} {\:\mathrm{lim}}\:\int_{\mathrm{1}} ^{\:\:{t}} \frac{\mathrm{1}}{{x}^{\mathrm{2}\:} }\:{dx}\:???? \\ $$

Question Number 66827 Answers: 0 Comments: 3

Question Number 66814 Answers: 0 Comments: 0

Let consider an integer serie {a_n x^n } given by a_n = H_n =Σ_(k=1) ^n (1/k) 1) Find out the largest domain D of convergence of that integer serie 2) ∀ x∈D , explicit the sum S(x) of the {a_n x^n } 3) Calculate ∫_(−1) ^1 S(1−x)S(x) dx .

$${Let}\:{consider}\:{an}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\:{given}\:{by}\:\:{a}_{{n}} \:=\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\: \\ $$$$\left.\mathrm{1}\right)\:{Find}\:{out}\:{the}\:{largest}\:{domain}\:{D}\:{of}\:{convergence}\:{of}\:{that}\:{integer}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:\forall\:{x}\in{D}\:\:,\:{explicit}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\: \\ $$$$\left.\mathrm{3}\right)\:{Calculate}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:{S}\left(\mathrm{1}−{x}\right){S}\left({x}\right)\:{dx}\:. \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 66803 Answers: 1 Comments: 3

prove that Σ_(r=k) ^n r = (1/2)n(n+1) show with a diagram that the volume of a parallepipe is a.(b×c)

$$\:{prove}\:{that} \\ $$$$\underset{{r}={k}} {\overset{{n}} {\sum}}\:{r}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}+\mathrm{1}\right) \\ $$$$ \\ $$$${show}\:{with}\:{a}\:{diagram}\:{that}\:{the}\:{volume}\:{of}\:{a}\:{parallepipe}\:{is}\:\:\:{a}.\left({b}×{c}\right) \\ $$

Question Number 66802 Answers: 0 Comments: 6

given that f(x) = 3x^3 − 2x^2 + 5x + 7 find a) α + β + γ b) αβγ c) α^2 + β^2 + γ^2 d) α^3 + β^3 + γ^3 any solutions directly?

$${given}\:{that}\: \\ $$$${f}\left({x}\right)\:=\:\mathrm{3}{x}^{\mathrm{3}} \:−\:\mathrm{2}{x}^{\mathrm{2}} \:+\:\mathrm{5}{x}\:+\:\mathrm{7}\:\:{find} \\ $$$$\left.{a}\right)\:\:\alpha\:+\:\beta\:+\:\gamma \\ $$$$\left.{b}\right)\:\alpha\beta\gamma\:\: \\ $$$$\left.{c}\right)\:\alpha^{\mathrm{2}} \:+\:\beta^{\mathrm{2}} \:+\:\gamma^{\mathrm{2}} \\ $$$$\left.{d}\right)\:\alpha^{\mathrm{3}} \:+\:\beta^{\mathrm{3}} \:+\:\gamma^{\mathrm{3}} \\ $$$${any}\:\:{solutions}\:\:{directly}? \\ $$

Question Number 66801 Answers: 0 Comments: 3

let f(x) =∫_0 ^2 (√(x+t^2 ))dt with x≥0 1) calculate f(x) 2)calculate g(x) =∫_0 ^2 (dt/(√(x+t^2 ))) 3)find the value[of ∫_0 ^2 (√(4+t^2 ))dt and ∫_0 ^2 (dt/(√(3+t^2 ))) 4) give g^′ (x) at form of integral.

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{{x}+{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \:\frac{{dt}}{\sqrt{{x}+{t}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\left[{of}\:\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{\mathrm{4}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{2}} \frac{{dt}}{\sqrt{\mathrm{3}+{t}^{\mathrm{2}} }}\right. \\ $$$$\left.\mathrm{4}\right)\:{give}\:{g}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}. \\ $$

Question Number 66800 Answers: 0 Comments: 1

calculate U_n =∫_(1/n) ^n ((arctan(x))/(1+x^2 ))dx and determine lim_(n→+∞) U_n 2)find nature of Σ U_n

$${calculate}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\:\:\frac{{arctan}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$${and}\:{determine}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 66799 Answers: 0 Comments: 0

let S_n =Σ_(k=1) ^n (((−1)^k )/(√k)) find a equivalent of S_n when n→+∞

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{{k}}}\:\:{find}\:{a}\:{equivalent}\:{of}\:{S}_{{n}} \:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 66798 Answers: 0 Comments: 0

find S_n =Σ_(k=0) ^n k^2 (C_n ^k )^2

$${find}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} \\ $$

Question Number 66797 Answers: 0 Comments: 0

find S_n =Σ_(k=0) ^n (C_n ^k )^3

$${find}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{3}} \\ $$

Question Number 66796 Answers: 0 Comments: 2

find the value of ∫_0 ^1 ((ln(1+x^2 ))/x^2 )dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 66795 Answers: 0 Comments: 3

let f(x) =e^(−x) ln(1+x^2 ) 1) calculate f^((n)) (0) 2) developp f at integr serie

$${let}\:{f}\left({x}\right)\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 66794 Answers: 0 Comments: 1

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

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

Question Number 66793 Answers: 0 Comments: 0

calculate ∫_0 ^1 cos(3arctanx)dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{3}{arctanx}\right){dx} \\ $$

Question Number 66792 Answers: 0 Comments: 1

calculate ∫_0 ^1 cos(2 arctan(x))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{2}\:{arctan}\left({x}\right)\right){dx} \\ $$

Question Number 66791 Answers: 0 Comments: 1

let f(x) =cos(2arctanx) 1) calculate f^((n)) (0) 2)developp f at integr serie

$${let}\:\:{f}\left({x}\right)\:={cos}\left(\mathrm{2}{arctanx}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 66790 Answers: 0 Comments: 0

find ∫_0 ^∞ (x/(sh(x)))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}}{{sh}\left({x}\right)}{dx} \\ $$

Question Number 66789 Answers: 0 Comments: 1

sove the (de) (1+2(√x))y^′ −(x+(√(x−1)))y =xsin(2x)

$${sove}\:{the}\:\left({de}\right)\:\:\:\left(\mathrm{1}+\mathrm{2}\sqrt{{x}}\right){y}^{'} −\left({x}+\sqrt{{x}−\mathrm{1}}\right){y}\:={xsin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 66788 Answers: 0 Comments: 0

solve the (de) (2x+1)y^′ +(x^2 −1)y =x^3 e^(−x)

$${solve}\:{the}\:\left({de}\right)\:\:\:\:\:\left(\mathrm{2}{x}+\mathrm{1}\right){y}^{'} \:\:\:+\left({x}^{\mathrm{2}} −\mathrm{1}\right){y}\:={x}^{\mathrm{3}} {e}^{−{x}} \\ $$

Question Number 66787 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ (x^2 /(ch(x)))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} }{{ch}\left({x}\right)}{dx} \\ $$

Question Number 66786 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (x/(ch(x)))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}}{{ch}\left({x}\right)}{dx} \\ $$

Question Number 66778 Answers: 1 Comments: 4

(1/1)+(1/2)−(2/3)+(1/4)+(1/5)−(2/6)+(1/7)+(1/8)−(2/9)+(1/(10))+(1/(11))−(2/(12))+∙∙∙=

$$\frac{\mathrm{1}}{\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{2}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}−\frac{\mathrm{2}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{7}}+\frac{\mathrm{1}}{\mathrm{8}}−\frac{\mathrm{2}}{\mathrm{9}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{11}}−\frac{\mathrm{2}}{\mathrm{12}}+\centerdot\centerdot\centerdot= \\ $$

Question Number 66769 Answers: 1 Comments: 2

simplify ((x+4)/(x−4))−((5x+20)/(x^2 −16))

$${simplify} \\ $$$$\frac{{x}+\mathrm{4}}{{x}−\mathrm{4}}−\frac{\mathrm{5}{x}+\mathrm{20}}{{x}^{\mathrm{2}} −\mathrm{16}} \\ $$

Question Number 66768 Answers: 0 Comments: 1

without using mathematical tables evaluate ((sin 60.tan 30.cos 60+sin 30.cos 45.sin 45)/(sin 90.cos 45.sin 45−sin 60.cos 30.sin 30))

$${without}\:{using}\:{mathematical}\:{tables} \\ $$$${evaluate} \\ $$$$\frac{\mathrm{sin}\:\mathrm{60}.\mathrm{tan}\:\mathrm{30}.\mathrm{cos}\:\mathrm{60}+\mathrm{sin}\:\mathrm{30}.\mathrm{cos}\:\mathrm{45}.\mathrm{sin}\:\mathrm{45}}{\mathrm{sin}\:\mathrm{90}.\mathrm{cos}\:\mathrm{45}.\mathrm{sin}\:\mathrm{45}−\mathrm{sin}\:\mathrm{60}.\mathrm{cos}\:\mathrm{30}.\mathrm{sin}\:\mathrm{30}} \\ $$

Question Number 66767 Answers: 0 Comments: 2

In a school there are 30 more boys than girls. One-quarter of the boys and two-thirds of the girls are boarders. If there are 255 boarders, find the number of students in the school.

$${In}\:{a}\:{school}\:{there}\:{are}\:\mathrm{30}\:{more}\:{boys} \\ $$$${than}\:{girls}.\:{One}-{quarter}\:{of}\:{the}\:{boys} \\ $$$${and}\:{two}-{thirds}\:{of}\:{the}\:{girls}\:{are}\:{boarders}. \\ $$$${If}\:{there}\:{are}\:\mathrm{255}\:{boarders},\:{find}\:{the} \\ $$$${number}\:{of}\:{students}\:{in}\:{the}\:{school}. \\ $$

Question Number 66765 Answers: 0 Comments: 2

Simba had 57 denomination notes which he deposited in his account. He had six times as many two-hundred shilling notes as one-thousand shilling notes and twice as many one-hundred shilling notes as two- hundred shilling notes. The rest were fifty shilling notes. If he deposited a total of sh 7750, find the number of fifty shilling notes he had.

$${Simba}\:{had}\:\mathrm{57}\:{denomination}\:{notes} \\ $$$${which}\:{he}\:{deposited}\:{in}\:{his}\:{account}. \\ $$$${He}\:{had}\:{six}\:{times}\:{as}\:{many}\:{two}-{hundred} \\ $$$${shilling}\:{notes}\:{as}\:{one}-{thousand}\: \\ $$$${shilling}\:{notes}\:{and}\:{twice}\:{as}\:{many}\: \\ $$$${one}-{hundred}\:{shilling}\:{notes}\:{as}\:{two}- \\ $$$${hundred}\:{shilling}\:{notes}.\:{The}\:{rest}\:{were} \\ $$$${fifty}\:{shilling}\:{notes}.\:{If}\:{he}\:{deposited} \\ $$$${a}\:{total}\:{of}\:{sh}\:\mathrm{7750},\:{find}\:{the}\:{number} \\ $$$${of}\:{fifty}\:{shilling}\:{notes}\:{he}\:{had}. \\ $$

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