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AllQuestion and Answers: Page 713

Question Number 146758 Answers: 2 Comments: 0

find forier series to half rang of f(x)=sinx ,0<x<π and prove that Σ_(n=1) ^∞ (1/(4n^2 −1))=(1/2)

$${find}\:{forier}\:{series}\:{to}\:{half}\:{rang}\:{of}\: \\ $$$${f}\left({x}\right)={sinx}\:\:,\mathrm{0}<{x}<\pi\:{and}\:{prove}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 146756 Answers: 2 Comments: 0

∫_( 0) ^( 1) t^2 + 1 dt

$$ \\ $$$$ \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{t}^{\mathrm{2}} \:+\:\mathrm{1}\:{dt} \\ $$

Question Number 146755 Answers: 0 Comments: 0

lim_(p→+∞) Σ_(k=1) ^(p−1) (2/(k^2 (p−k)^2 ))=...?

$$\underset{{p}\rightarrow+\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{p}−\mathrm{1}} {\sum}}\frac{\mathrm{2}}{{k}^{\mathrm{2}} \left({p}−{k}\right)^{\mathrm{2}} }=...? \\ $$

Question Number 146752 Answers: 1 Comments: 0

∫_( 0) ^( 1) t^2 + (1/2)t −6dx

$$ \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{t}^{\mathrm{2}} +\:\frac{\mathrm{1}}{\mathrm{2}}{t}\:−\mathrm{6}{dx}\:\: \\ $$

Question Number 146746 Answers: 4 Comments: 0

∫_( 0) ^( 4) (√(16 - x^2 )) dx = ?

$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{4}} {\int}}\:\sqrt{\mathrm{16}\:-\:{x}^{\mathrm{2}} }\:{dx}\:=\:? \\ $$

Question Number 146744 Answers: 1 Comments: 0

if ∫_a ^b f(x)dx = 7 and ∫_a ^b 4 g(x)dx = −6 find ∫_a ^b (3 f(x)−8 g(x)) dx = ?

$${if}\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}{f}\left({x}\right){dx}\:=\:\mathrm{7}\:\:\:{and}\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}\mathrm{4}\:{g}\left({x}\right){dx}\:=\:−\mathrm{6} \\ $$$${find}\:\:\:\underset{\boldsymbol{{a}}} {\overset{\boldsymbol{{b}}} {\int}}\left(\mathrm{3}\:{f}\left({x}\right)−\mathrm{8}\:{g}\left({x}\right)\right)\:{dx}\:=\:? \\ $$

Question Number 146736 Answers: 1 Comments: 0

1: S:= Σ_(n=1) ^∞ (((−1)^( n−1) )/(n.2^( n) )) =? 2: A:= Σ(((−1)^( n−1) )/(n^2 . 2^( n) )) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{1}:\:\:\:\:\mathrm{S}:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\:{n}−\mathrm{1}} }{{n}.\mathrm{2}^{\:{n}} }\:=? \\ $$$$\:\:\:\:\:\:\mathrm{2}:\:\:\:\:\mathrm{A}:=\:\Sigma\frac{\left(−\mathrm{1}\right)^{\:{n}−\mathrm{1}} }{{n}^{\mathrm{2}} .\:\mathrm{2}^{\:{n}} }\:=? \\ $$

Question Number 146735 Answers: 2 Comments: 0

Σ_(n=1) ^∞ ((1+(1/2)+(1/3)+...+(1/n))/((n+1)(n+2)))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+...+\frac{\mathrm{1}}{\mathrm{n}}}{\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{2}\right)}=? \\ $$

Question Number 146729 Answers: 0 Comments: 0

Question Number 146725 Answers: 2 Comments: 0

cos(x) ∙ cos(3x) = cos(5x) ∙ cos(7x) ⇒ x = ?

$${cos}\left({x}\right)\:\centerdot\:{cos}\left(\mathrm{3}{x}\right)\:=\:{cos}\left(\mathrm{5}{x}\right)\:\centerdot\:{cos}\left(\mathrm{7}{x}\right) \\ $$$$\Rightarrow\:{x}\:=\:? \\ $$

Question Number 146705 Answers: 1 Comments: 0

find lim_(x→0) ((sin(tan(2x)−x)+1−cos(πx^2 ))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{sin}\left(\mathrm{tan}\left(\mathrm{2x}\right)−\mathrm{x}\right)+\mathrm{1}−\mathrm{cos}\left(\pi\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 146702 Answers: 1 Comments: 0

Question Number 146701 Answers: 3 Comments: 2

Question Number 146697 Answers: 0 Comments: 0

∀t≥−1,F(t)=(2/π)∫_0 ^(π/2) (√(1+tcos^2 ϕ))dϕ 1) Show that ∀t≤−1 F(t)=(√(1+t))F(−(1/(1+t))) 2) show that if 0≤t_1 , 0≤F(t_2 )−F(t_1 )≤((t_2 −t_1 )/4)

$$\forall{t}\geqslant−\mathrm{1},{F}\left({t}\right)=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{1}+{tcos}^{\mathrm{2}} \varphi}{d}\varphi \\ $$$$\left.\mathrm{1}\right)\:{Show}\:{that}\:\forall{t}\leqslant−\mathrm{1}\:{F}\left({t}\right)=\sqrt{\mathrm{1}+{t}}{F}\left(−\frac{\mathrm{1}}{\mathrm{1}+{t}}\right) \\ $$$$\left.\mathrm{2}\right)\:{show}\:{that}\:{if}\:\mathrm{0}\leqslant{t}_{\mathrm{1}} \:, \\ $$$$\mathrm{0}\leqslant{F}\left({t}_{\mathrm{2}} \right)−{F}\left({t}_{\mathrm{1}} \right)\leqslant\frac{{t}_{\mathrm{2}} −{t}_{\mathrm{1}} }{\mathrm{4}} \\ $$

Question Number 146689 Answers: 1 Comments: 0

∫ ((−3x+5)/( (((3x^2 −10x)^2 ))^(1/3) )) dx = ... ? Solve it without substitution method.

$$\int\:\:\frac{−\mathrm{3}{x}+\mathrm{5}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{10}{x}\right)^{\mathrm{2}} }}\:\:{dx}\:\:=\:\:\:...\:\:? \\ $$$${Solve}\:\:{it}\:\:{without}\:\:{substitution}\:\:{method}. \\ $$

Question Number 146687 Answers: 0 Comments: 0

E = MC^(2 ) ∫requency = E/M

$${E}\:=\:{MC}^{\mathrm{2}\:} \:\:\:\:\:\:\:\:\:\int{requency}\:=\:{E}/{M} \\ $$

Question Number 146680 Answers: 2 Comments: 0

Compare: 100^(101) and 101^(100)

$${Compare}:\:\:\mathrm{100}^{\mathrm{101}} \:\:{and}\:\:\:\mathrm{101}^{\mathrm{100}} \\ $$

Question Number 146678 Answers: 2 Comments: 0

∫ (√(2 + x^2 )) dx = ?

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

Question Number 146677 Answers: 1 Comments: 0

if f(x) = 3^(x+1) find ((f(2x + 1))/(f(x + 1))) = ?

$${if}\:\:\:{f}\left({x}\right)\:=\:\mathrm{3}^{\boldsymbol{{x}}+\mathrm{1}} \:\:\:{find}\:\:\:\frac{{f}\left(\mathrm{2}{x}\:+\:\mathrm{1}\right)}{{f}\left({x}\:+\:\mathrm{1}\right)}\:=\:? \\ $$

Question Number 146672 Answers: 1 Comments: 0

Question Number 146669 Answers: 1 Comments: 0

∫_0 ^π (a−e^(−ix) )^n (a−e^(ix) )^n cos(nx)dx

$$\int_{\mathrm{0}} ^{\pi} \left(\mathrm{a}−\mathrm{e}^{−\mathrm{ix}} \right)^{\mathrm{n}} \left(\mathrm{a}−\mathrm{e}^{\mathrm{ix}} \right)^{\mathrm{n}} \mathrm{cos}\left(\mathrm{nx}\right)\mathrm{dx} \\ $$

Question Number 146667 Answers: 1 Comments: 0

Question Number 146730 Answers: 1 Comments: 2

Question Number 146653 Answers: 1 Comments: 0

Question Number 146722 Answers: 1 Comments: 0

D=lim_(n→+∝) (n^2 /(n−1))[((sin((88)/n))/(1+2)) + ((sin((88)/n))/(1+2+3)) + ...+ ((sin((88)/n))/(1+2+3+...+n))]

$$\mathrm{D}=\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{n}−\mathrm{1}}\left[\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}}\:+\:\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}\:+\:...+\:\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{n}}\right] \\ $$

Question Number 146636 Answers: 1 Comments: 2

4 + ((12)/(8 + (7/(3 + (4/(x + 1)))))) = 8 ⇒ x=?

$$\mathrm{4}\:+\:\frac{\mathrm{12}}{\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}}\:=\:\mathrm{8}\:\:\Rightarrow\:\:\boldsymbol{{x}}=? \\ $$

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