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Question Number 106825 Answers: 5 Comments: 0

lim_(x→0) ((sin5x − tan5x)/x^3 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\mathrm{5}{x}\:−\:{tan}\mathrm{5}{x}}{{x}^{\mathrm{3}} } \\ $$

Question Number 106816 Answers: 2 Comments: 1

prove by mathematical induction (1) n^2 ≤ 2^n ; n ≥ 4 (2) (n+1)^2 < 2n^2 ; n ≥ 3 (3) 2^n −3 ≥ 2^(n−2) ; n ≥ 5 @bemath@

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{n}^{\mathrm{2}} \:\leqslant\:\mathrm{2}^{\mathrm{n}} \:;\:\mathrm{n}\:\geqslant\:\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{2}} \:<\:\mathrm{2n}^{\mathrm{2}} \:;\:\mathrm{n}\:\geqslant\:\mathrm{3} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}^{\mathrm{n}} −\mathrm{3}\:\geqslant\:\mathrm{2}^{\mathrm{n}−\mathrm{2}} \:;\:\mathrm{n}\:\geqslant\:\mathrm{5} \\ $$$$\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$

Question Number 106815 Answers: 1 Comments: 0

please help me to show that the equation X^n +aX+c=0 can not have more than 3 reals solutions

$$\:\:\:\boldsymbol{{please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}\:\boldsymbol{{to}}\:\boldsymbol{{show}} \\ $$$$\boldsymbol{{that}}\:\:\boldsymbol{{the}}\:\boldsymbol{{equation}}\: \\ $$$$\:\boldsymbol{{X}}^{\boldsymbol{{n}}} +\boldsymbol{{aX}}+\boldsymbol{{c}}=\mathrm{0}\:\boldsymbol{{can}}\:\boldsymbol{{not}}\:\boldsymbol{{have}} \\ $$$$\boldsymbol{{more}}\:\boldsymbol{{than}}\:\mathrm{3}\:\boldsymbol{{reals}}\:\boldsymbol{{solutions}} \\ $$$$ \\ $$

Question Number 106810 Answers: 3 Comments: 0

lim_(x→0) ((tanx−sinx)/(sinx(cos2x−cosx)))=???

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{{tanx}−{sinx}}{{sinx}\left({cos}\mathrm{2}{x}−{cosx}\right)}=??? \\ $$

Question Number 106809 Answers: 2 Comments: 0

∫e^(2x) sine^x dx=??

$$\int{e}^{\mathrm{2}{x}} {sine}^{{x}} {dx}=?? \\ $$

Question Number 106808 Answers: 0 Comments: 0

∫tan(lnx)dx=???

$$\int{tan}\left({lnx}\right){dx}=??? \\ $$

Question Number 106807 Answers: 1 Comments: 0

∫sin(ln3x)dx=???

$$\int{sin}\left({ln}\mathrm{3}{x}\right){dx}=??? \\ $$

Question Number 106794 Answers: 3 Comments: 0

repost old question unanswer Given → { ((((4x^2 )/(1+4x^2 )) = y)),((((4y^2 )/(1+4y^2 )) = z)),((((4z^2 )/(1+4z^2 )) = x)) :}

$$\mathrm{repost}\:\mathrm{old}\:\mathrm{question}\:\mathrm{unanswer} \\ $$$$\mathcal{G}\mathrm{iven}\:\rightarrow\begin{cases}{\frac{\mathrm{4x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4x}^{\mathrm{2}} }\:=\:\mathrm{y}}\\{\frac{\mathrm{4y}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4y}^{\mathrm{2}} }\:=\:\mathrm{z}}\\{\frac{\mathrm{4z}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4z}^{\mathrm{2}} }\:=\:\mathrm{x}}\end{cases} \\ $$

Question Number 106792 Answers: 1 Comments: 0

≻bobhans≺ From a batch containing 6 boys and 4 girls a group of 4 students is tobe selected . How many group formations will have exactly 2 girls?

$$\:\:\:\:\:\:\succ\mathrm{bobhans}\prec \\ $$$$\mathrm{From}\:\mathrm{a}\:\mathrm{batch}\:\mathrm{containing}\:\mathrm{6}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{4}\:\mathrm{girls} \\ $$$$\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{4}\:\mathrm{students}\:\mathrm{is}\:\mathrm{tobe}\:\mathrm{selected}\:. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{group}\:\mathrm{formations}\:\mathrm{will}\:\mathrm{have} \\ $$$$\mathrm{exactly}\:\mathrm{2}\:\mathrm{girls}? \\ $$

Question Number 106787 Answers: 2 Comments: 0

^(@bemath@) y = (cos 2x)^(sin x) , find (dy/dx) ?

$$\:\:\:\:\:\:\overset{@\mathrm{bemath}@} {\:} \\ $$$$\:\:\mathrm{y}\:=\:\left(\mathrm{cos}\:\mathrm{2x}\right)^{\mathrm{sin}\:\mathrm{x}} \:,\:\mathrm{find}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:? \\ $$

Question Number 106782 Answers: 1 Comments: 0

∫2xdx

$$\int\mathrm{2}{xdx} \\ $$

Question Number 106779 Answers: 3 Comments: 0

^(≻bobhans≺) How do you find a point on the curve y=x^2 closest to the point (0,18) ?

$$\:\:\:\:\:\:\:\:\:^{\succ\mathrm{bobhans}\prec} \\ $$$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{find}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} \\ $$$$\mathrm{closest}\:\mathrm{to}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{0},\mathrm{18}\right)\:? \\ $$

Question Number 106775 Answers: 3 Comments: 0

^(@bemath@) (1) (d^2 y/dx^2 ) −6(dy/dx) + 9y = 1+x+x^2 (2) { ((x^3 +3y^3 = 11)),((x^2 y +xy^2 = 6)) :}

$$\:\:\:\:\:\:\:\:\:\:\:^{@\mathrm{bemath}@} \\ $$$$\:\left(\mathrm{1}\right)\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:−\mathrm{6}\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\mathrm{9y}\:=\:\mathrm{1}+\mathrm{x}+\mathrm{x}^{\mathrm{2}} \\ $$$$\:\:\left(\mathrm{2}\right)\:\begin{cases}{\mathrm{x}^{\mathrm{3}} +\mathrm{3y}^{\mathrm{3}} \:=\:\mathrm{11}}\\{\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\mathrm{xy}^{\mathrm{2}} \:=\:\mathrm{6}}\end{cases}\: \\ $$

Question Number 106771 Answers: 2 Comments: 1

@bemath@ lim_(x→π) [((4(x−π) cos^2 x)/(π(π−2x) cos (x−(π/2))))]= ?

$$\:\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\left[\frac{\mathrm{4}\left(\mathrm{x}−\pi\right)\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}{\pi\left(\pi−\mathrm{2x}\right)\:\mathrm{cos}\:\left(\mathrm{x}−\frac{\pi}{\mathrm{2}}\right)}\right]=\:? \\ $$

Question Number 106745 Answers: 2 Comments: 8

Question Number 106744 Answers: 0 Comments: 3

∫ x^x^x^x dx

$$\int\:{x}^{{x}^{{x}^{{x}} } } {dx} \\ $$

Question Number 106743 Answers: 1 Comments: 0

∫(√(secy))dy

$$\int\sqrt{{secy}}{dy} \\ $$

Question Number 106730 Answers: 1 Comments: 0

Question Number 106727 Answers: 2 Comments: 0

Question Number 106726 Answers: 1 Comments: 0

Prove sin5θ=16sin^5 θ−20sin^3 θ+5sinθ Hence, show that sin 6° is an irrational number.

$$\mathrm{Prove}\:\mathrm{sin5}\theta=\mathrm{16sin}^{\mathrm{5}} \theta−\mathrm{20sin}^{\mathrm{3}} \theta+\mathrm{5sin}\theta \\ $$$$\mathrm{Hence},\:\mathrm{show}\:\mathrm{that}\:\mathrm{sin}\:\mathrm{6}°\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{irrational}\:\mathrm{number}.\: \\ $$

Question Number 106774 Answers: 4 Comments: 0

^(@bemath@) (1) 3^x + 3^((√x) ) = 90. find x ? (2) x (dy/dx)−(1+x)y = xy^2

$$\:\:\:\:\overset{@\mathrm{bemath}@} {\:} \\ $$$$\:\left(\mathrm{1}\right)\:\:\:\:\mathrm{3}^{{x}} \:+\:\mathrm{3}^{\sqrt{{x}}\:} =\:\mathrm{90}.\:\mathrm{find}\:{x}\:?\: \\ $$$$\:\:\left(\mathrm{2}\right)\:\mathrm{x}\:\frac{\mathrm{dy}}{\mathrm{dx}}−\left(\mathrm{1}+\mathrm{x}\right)\mathrm{y}\:=\:\mathrm{xy}^{\mathrm{2}} \\ $$

Question Number 106709 Answers: 3 Comments: 0

∫ ((sin^2 x dx)/(1−sin x cos x)) ?

$$ \\ $$$$\:\int\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}}{\mathrm{1}−\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}\:? \\ $$

Question Number 106707 Answers: 2 Comments: 0

@bemath@ Given { ((f(x)=2x+3)),(((g○f)(x)=2x−1)) :} find (f○g)(2).

$$\:\:\:\:@\mathrm{bemath}@ \\ $$$$\mathcal{G}\mathrm{iven}\:\begin{cases}{\mathrm{f}\left(\mathrm{x}\right)=\mathrm{2x}+\mathrm{3}}\\{\left(\mathrm{g}\circ\mathrm{f}\right)\left(\mathrm{x}\right)=\mathrm{2x}−\mathrm{1}}\end{cases} \\ $$$$\mathrm{find}\:\left(\mathrm{f}\circ\mathrm{g}\right)\left(\mathrm{2}\right). \\ $$

Question Number 106705 Answers: 1 Comments: 0

@bemath@ ∫ ((1+x^2 )/(√(9−4x^2 ))) dx

$$\:\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$$$\:\:\:\:\:\int\:\frac{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }{\sqrt{\mathrm{9}−\mathrm{4x}^{\mathrm{2}} }}\:\mathrm{dx}\: \\ $$

Question Number 106695 Answers: 1 Comments: 0

#bobhans# 3^x −2^(x+1) ≤ (√(2.9^x −10.6^x +2^(2x+3) )) find the solution set

$$\:\:\:\:#\mathrm{bobhans}# \\ $$$$\mathrm{3}^{\mathrm{x}} −\mathrm{2}^{\mathrm{x}+\mathrm{1}} \:\leqslant\:\sqrt{\mathrm{2}.\mathrm{9}^{\mathrm{x}} −\mathrm{10}.\mathrm{6}^{\mathrm{x}} +\mathrm{2}^{\mathrm{2x}+\mathrm{3}} } \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set} \\ $$

Question Number 106694 Answers: 3 Comments: 0

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