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Question Number 120729 Answers: 1 Comments: 0

Question Number 120727 Answers: 1 Comments: 0

Question Number 120724 Answers: 1 Comments: 2

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Question Number 120720 Answers: 2 Comments: 0

Given lim_(x→∞) (√(x+2(√x)+3)) −(√x)+b = 4 find the value of b^2 +1

$$\:\mathrm{Given}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{x}+\mathrm{2}\sqrt{\mathrm{x}}+\mathrm{3}}\:−\sqrt{\mathrm{x}}+\mathrm{b}\:=\:\mathrm{4} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{b}^{\mathrm{2}} +\mathrm{1} \\ $$

Question Number 120715 Answers: 0 Comments: 0

Question Number 120711 Answers: 3 Comments: 0

show that 1+2+3+4...=((−1)/8)

$${show}\:{that}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}...=\frac{−\mathrm{1}}{\mathrm{8}} \\ $$

Question Number 120706 Answers: 0 Comments: 1

If f(x)=x^4 +ax^3 +bx^2 +cx+d f(1)=5, f(2)=10, f(3)=15 find f(9)+f(−5).

$$\mathrm{If}\:{f}\left({x}\right)={x}^{\mathrm{4}} +{ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{5},\:{f}\left(\mathrm{2}\right)=\mathrm{10},\:{f}\left(\mathrm{3}\right)=\mathrm{15} \\ $$$$\mathrm{find}\:{f}\left(\mathrm{9}\right)+{f}\left(−\mathrm{5}\right). \\ $$

Question Number 120705 Answers: 0 Comments: 1

Given that the curve y=2x^2 −19x+18 does not intersect the line y=x+k, find the largest integer of k.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{curve}\:{y}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{19}{x}+\mathrm{18} \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{intersect}\:\mathrm{the}\:\mathrm{line}\:{y}={x}+{k}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{integer}\:\mathrm{of}\:{k}. \\ $$

Question Number 120703 Answers: 1 Comments: 0

∫ (dx/(x^2 (√(9+x^2 )))) ?

$$\:\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} \:\sqrt{\mathrm{9}+\mathrm{x}^{\mathrm{2}} }}\:? \\ $$

Question Number 120702 Answers: 0 Comments: 0

Question Number 120688 Answers: 1 Comments: 0

∫_0 ^π ((x dx)/(1+sin^2 x))

$$\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{x}\:\mathrm{dx}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}\: \\ $$$$ \\ $$

Question Number 120686 Answers: 3 Comments: 0

lim_(x→0) ((sin x−tan x)/x^3 ) ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{tan}\:\mathrm{x}}{\mathrm{x}^{\mathrm{3}} }\:? \\ $$

Question Number 120679 Answers: 2 Comments: 3

$${h} \\ $$

Question Number 120676 Answers: 0 Comments: 4

Question Number 120654 Answers: 2 Comments: 0

∫_0 ^∞ (dx/([x+(√(1+x^2 )) ]^n ))

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\left[\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\right]^{\mathrm{n}} } \\ $$

Question Number 120636 Answers: 0 Comments: 23

selective Binomial

$${selective}\:{Binomial} \\ $$

Question Number 120631 Answers: 1 Comments: 0

Question Number 120628 Answers: 0 Comments: 15

selective intregals

$${selective}\:{intregals} \\ $$

Question Number 120621 Answers: 0 Comments: 8

Trigonometry selective questions

$${Trigonometry} \\ $$$${selective}\:{questions} \\ $$

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Question Number 120614 Answers: 0 Comments: 8

Trigonometry

$${Trigonometry}\: \\ $$

Question Number 120601 Answers: 1 Comments: 0

Find the range of values of x for which the expansion of the binomial (2 − 3x)^(−4) is valid. I need help with explanation please

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{for}\: \\ $$$$\:\mathrm{which}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\:\left(\mathrm{2}\:−\:\mathrm{3}{x}\right)^{−\mathrm{4}} \:\mathrm{is}\:\mathrm{valid}.\: \\ $$$$\:{I}\:{need}\:{help}\:{with}\:{explanation}\:{please} \\ $$

Question Number 120599 Answers: 0 Comments: 0

Let x = u^6 dx = 6u^5 du I = ∫(u^3 /((1+u^2 )^2 )) ×6u^5 du =6 ∫(u^8 /(1+2u^2 +u^4 )) du =6 ∫[((−4)/(u^2 +1))+(1/((1+u^2 )^2 ))+u^4 −2u^2 +3]du =6[−4tan^(−1) (u) + I_1 + (u^5 /5) − (2/3)u^3 +3u]+c I_1 = ∫(1/((1+u^2 )^2 )) du Put u = tan z du = sec^2 z dz I_1 = ∫(1/(sec^4 z))×sec^2 z dz = ∫cos^2 z dz = ∫(1/2)(cos 2z + 1)dz = (1/2)((1/2) sin 2z + z) = (1/2)×(u/(1+u^2 )) + (1/2) tan^(−1) u I = −4 tan^(−1) u + (u/(2(1+u^2 ))) + (1/2) tan^(−1) u +(u^5 /5) − (2/3)u^3 + 3u + c = −(7/2) tan^(−1) u + (u/(2(1+u^2 ))) + (1/5)u^5 + 3u + c = −(7/2) tan^(−1) ( ^6 (√x)) + ((x)^(1/6) /(2(1+(x^2 )^(1/6) ))) + (1/5) (x^5 )^(1/6) + 3 (x)^(1/6) + c

$${Let}\:{x}\:=\:{u}^{\mathrm{6}} \:\:\:\:\:\:\:\:\:{dx}\:=\:\mathrm{6}{u}^{\mathrm{5}} \:{du} \\ $$$${I}\:=\:\int\frac{{u}^{\mathrm{3}} }{\left(\mathrm{1}+{u}^{\mathrm{2}} \right)^{\mathrm{2}} }\:×\mathrm{6}{u}^{\mathrm{5}} \:{du} \\ $$$$\:\:\:=\mathrm{6}\:\int\frac{{u}^{\mathrm{8}} }{\mathrm{1}+\mathrm{2}{u}^{\mathrm{2}} +{u}^{\mathrm{4}} }\:{du} \\ $$$$\:\:\:=\mathrm{6}\:\int\left[\frac{−\mathrm{4}}{{u}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{1}}{\left(\mathrm{1}+{u}^{\mathrm{2}} \right)^{\mathrm{2}} }+{u}^{\mathrm{4}} −\mathrm{2}{u}^{\mathrm{2}} +\mathrm{3}\right]{du} \\ $$$$\:\:\:=\mathrm{6}\left[−\mathrm{4}{tan}^{−\mathrm{1}} \left({u}\right)\:+\:{I}_{\mathrm{1}} \:+\:\frac{{u}^{\mathrm{5}} }{\mathrm{5}}\:−\:\frac{\mathrm{2}}{\mathrm{3}}{u}^{\mathrm{3}} \:+\mathrm{3}{u}\right]+{c} \\ $$$${I}_{\mathrm{1}} \:=\:\int\frac{\mathrm{1}}{\left(\mathrm{1}+{u}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{du} \\ $$$${Put}\:{u}\:=\:{tan}\:{z}\:\:\:\:\:{du}\:=\:{sec}^{\mathrm{2}} {z}\:{dz} \\ $$$${I}_{\mathrm{1}} \:=\:\int\frac{\mathrm{1}}{{sec}^{\mathrm{4}} {z}}×{sec}^{\mathrm{2}} {z}\:{dz}\:=\:\int{cos}^{\mathrm{2}} {z}\:{dz} \\ $$$$\:\:\:\:\:=\:\int\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\:\mathrm{2}{z}\:+\:\mathrm{1}\right){dz} \\ $$$$\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}\:{sin}\:\mathrm{2}{z}\:+\:{z}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}×\frac{{u}}{\mathrm{1}+{u}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} {u} \\ $$$${I}\:=\:−\mathrm{4}\:{tan}^{−\mathrm{1}} {u}\:+\:\frac{{u}}{\mathrm{2}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} {u}\:+\frac{{u}^{\mathrm{5}} }{\mathrm{5}}\:−\:\frac{\mathrm{2}}{\mathrm{3}}{u}^{\mathrm{3}} \:+\:\mathrm{3}{u}\:+\:{c} \\ $$$$\:\:\:=\:−\frac{\mathrm{7}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} {u}\:+\:\frac{{u}}{\mathrm{2}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}\:+\:\frac{\mathrm{1}}{\mathrm{5}}{u}^{\mathrm{5}} \:+\:\mathrm{3}{u}\:+\:{c} \\ $$$$\:\:\:=\:−\frac{\mathrm{7}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} \left(\overset{\mathrm{6}} {\:}\sqrt{{x}}\right)\:+\:\frac{\sqrt[{\mathrm{6}}]{{x}}}{\mathrm{2}\left(\mathrm{1}+\sqrt[{\mathrm{6}}]{{x}^{\mathrm{2}} }\right)}\:+\:\frac{\mathrm{1}}{\mathrm{5}}\:\sqrt[{\mathrm{6}}]{{x}^{\mathrm{5}} }\:+\:\mathrm{3}\:\sqrt[{\mathrm{6}}]{{x}}\:+\:{c} \\ $$

Question Number 120598 Answers: 0 Comments: 0

Let A be a subset of a Real number with dimension 2 and let x be a real number member of dimension 2. Then x is called the limit point of A if ...?

$$ \\ $$$$\mathrm{Let}\:\mathrm{A}\:\mathrm{be}\:\mathrm{a}\:\mathrm{subset}\:\mathrm{of}\:\mathrm{a}\:\mathrm{Real}\:\mathrm{number} \\ $$$$\mathrm{wit}{h}\:\mathrm{dimension}\:\mathrm{2}\:\mathrm{and}\:\mathrm{let}\:\mathrm{x}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real} \\ $$$$\mathrm{num}{b}\mathrm{er}\:\mathrm{member}\:\mathrm{of}\:\mathrm{dimension}\:\mathrm{2}.\:\mathrm{Then}\: \\ $$$$\mathrm{x}\:\mathrm{is}\:\mathrm{called}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{point}\:\mathrm{of}\:\mathrm{A}\:\mathrm{if}\:...? \\ $$

Question Number 120597 Answers: 0 Comments: 0

Let A ≠ −φ , A are subsets of R and A are open sets. Then each p member A applies ...?

$$ \\ $$$$\mathrm{Let}\:\mathrm{A}\:\:\neq\:−\phi\:,\:\mathrm{A}\:\mathrm{are}\:\mathrm{subsets}\:\mathrm{of}\:\mathrm{R}\:\mathrm{and}\:\mathrm{A}\:\mathrm{are} \\ $$$$\mathrm{ope}{n}\:\mathrm{sets}.\:\mathrm{Then}\:\mathrm{each}\:\mathrm{p}\:\mathrm{member}\:\mathrm{A} \\ $$$$\mathrm{applie}{s}\:...? \\ $$

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