Question and Answers Forum

All Questions Topic List

IntegrationQuestion and Answers: Page 77

Question Number 146767 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (((−1)^n )/(n+1))(1+(1/3)+...+(1/(2n+1)))=?

$$\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}+\mathrm{1}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+...+\frac{\mathrm{1}}{\mathrm{2n}+\mathrm{1}}\right)=? \\ $$

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 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 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 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 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 146619 Answers: 1 Comments: 0

solve :: ( x ∈ R ) [ x ] = [ x^( 2) − x −6 ] note:: [x ] := max { q ∈ Z ∣ q ≤ x }

$$ \\ $$$$\:\:\:\:{solve}\:::\:\:\:\left(\:{x}\:\in\:\mathbb{R}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\:{x}\:\right]\:=\:\left[\:{x}^{\:\mathrm{2}} −\:{x}\:−\mathrm{6}\:\right] \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{note}::\:\:\:\left[{x}\:\right]\::=\:{max}\:\left\{\:{q}\:\in\:\mathbb{Z}\:\mid\:{q}\:\leqslant\:{x}\:\right\} \\ $$

Question Number 146597 Answers: 5 Comments: 0

∫ ((cos 5x+cos 4x)/(1−2cos 3x)) dx =? ∫ ((√(tan x))/(sin 2x)) dx =? ∫ (dx/( (√(cos^3 x sin^5 x)))) =?

$$\:\:\:\:\int\:\frac{\mathrm{cos}\:\mathrm{5x}+\mathrm{cos}\:\mathrm{4x}}{\mathrm{1}−\mathrm{2cos}\:\mathrm{3x}}\:\mathrm{dx}\:=?\: \\ $$$$\:\:\int\:\frac{\sqrt{\mathrm{tan}\:\mathrm{x}}}{\mathrm{sin}\:\mathrm{2x}}\:\mathrm{dx}\:=? \\ $$$$\:\:\int\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{cos}\:^{\mathrm{3}} \mathrm{x}\:\mathrm{sin}\:^{\mathrm{5}} \mathrm{x}}}\:=? \\ $$

Question Number 146584 Answers: 0 Comments: 0

......nice ... ... ... calculus...... Ω= ∫_0 ^( 4) x^( 2) d (⌊x+⌊x +⌊x+⌊x⌋⌋⌋)=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:......{nice}\:...\:...\:...\:{calculus}...... \\ $$$$\: \\ $$$$\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{4}} {x}^{\:\mathrm{2}} \:{d}\:\left(\lfloor{x}+\lfloor{x}\:+\lfloor{x}+\lfloor{x}\rfloor\rfloor\rfloor\right)=?\: \\ $$$$ \\ $$

Question Number 146582 Answers: 1 Comments: 0

∫ tan^4 x cos^2 x dx =?

$$\:\:\:\int\:\mathrm{tan}\:^{\mathrm{4}} \mathrm{x}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:=? \\ $$

Question Number 146577 Answers: 2 Comments: 1

Question Number 146455 Answers: 1 Comments: 0

Given that F(x) = ∫_0 ^x (t^2 /( (√(t^2 +1))))dt Show that F(x) is an increasing function

$$\mathrm{Given}\:\mathrm{that}\:\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{{x}} \frac{{t}^{\mathrm{2}} }{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}\: \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:{F}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{function} \\ $$

Question Number 146429 Answers: 1 Comments: 1

Σ_(n=1) ^∞ (n∙ln((2n+1)/(2n−1))−1)=?

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

Question Number 146361 Answers: 2 Comments: 0

find ∫(1/(x^n +1))dx for n∈N

$${find}\:\int\frac{\mathrm{1}}{{x}^{{n}} +\mathrm{1}}{dx}\:{for}\:{n}\in{N} \\ $$

Question Number 146307 Answers: 1 Comments: 0

S_n =Σ_(k=1) ^n (( 1)/(k (k+2)k+4))) lim_( n→∞) ( S_( n) ) = ?

$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{S}_{{n}} \:=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\:\mathrm{1}}{\left.{k}\:\left({k}+\mathrm{2}\right){k}+\mathrm{4}\right)} \\ $$$$\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\infty} \:\left(\:\:\mathrm{S}_{\:{n}} \:\right)\:=\:? \\ $$

Question Number 146290 Answers: 0 Comments: 5

Question Number 146181 Answers: 4 Comments: 0

Σ_(n=0) ^∞ (1/(2^( n) (n+1 ) ( n + 2 ))) =?

$$ \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{\:{n}} \:\left({n}+\mathrm{1}\:\right)\:\left(\:{n}\:+\:\mathrm{2}\:\right)}\:=? \\ $$

Question Number 146155 Answers: 0 Comments: 2

lim_(n→∞) (Arcsin(x))^( n) =0 ∴ x ∈ ? Q : mr liberty

$$ \\ $$$$\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \:\left({Arcsin}\left({x}\right)\right)^{\:{n}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\:\:\:\:{x}\:\in\:?\: \\ $$$$\:\:\:\:\:\:{Q}\::\:{mr}\:{liberty} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 146147 Answers: 2 Comments: 0

Υ = ∫ (dx/(x^4 (√(x^2 −a^2 )))) =?

$$\:\Upsilon\:=\:\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{4}} \:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} }}\:=? \\ $$

Question Number 146110 Answers: 2 Comments: 0

∫((x+1)/(2x^2 +x+1))dx

$$\int\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 146108 Answers: 0 Comments: 0

Solve in Z[X] 1) XP ′ ≡ −1 mod(X^4 +1) 2) X^3 P −P ′ ≡ 1−X^2 mod(X^4 +1) 3) P^2 −X^3 P−X^2 ≡ 0 mod(X^2 +2)

$$\:{Solve}\:\:{in}\:\mathbb{Z}\left[{X}\right] \\ $$$$\left.\mathrm{1}\right)\:{XP}\:'\:\equiv\:−\mathrm{1}\:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:{X}^{\mathrm{3}} {P}\:−{P}\:'\:\equiv\:\mathrm{1}−{X}^{\mathrm{2}} \:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{P}\:^{\mathrm{2}} −{X}^{\mathrm{3}} {P}−{X}^{\mathrm{2}} \:\:\equiv\:\mathrm{0}\:{mod}\left({X}^{\mathrm{2}} +\mathrm{2}\right) \\ $$

Question Number 146131 Answers: 1 Comments: 1

Question Number 146090 Answers: 2 Comments: 0

∫_0 ^∞ ((sinh(at)sinh(bt))/(sinh(ct)e^(tz) ))dt= ((ab)/(c(z^2 +c^2 −a^2 −b^2 +K_(k=1) ^∞ ((−4k^2 (k^2 c^2 −a^2 )(k^2 c^2 −b^2 ))/((2k+1)(z^2 +(2k^2 +2k+1)c^2 −a^2 −b^2 ))))))

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sinh}\left(\mathrm{at}\right)\mathrm{sinh}\left(\mathrm{bt}\right)}{\mathrm{sinh}\left(\mathrm{ct}\right)\mathrm{e}^{\mathrm{tz}} }\mathrm{dt}= \\ $$$$\frac{\mathrm{ab}}{\mathrm{c}\left(\mathrm{z}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} +\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\mathrm{K}}}\frac{−\mathrm{4k}^{\mathrm{2}} \left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \right)}{\left(\mathrm{2k}+\mathrm{1}\right)\left(\mathrm{z}^{\mathrm{2}} +\left(\mathrm{2k}^{\mathrm{2}} +\mathrm{2k}+\mathrm{1}\right)\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \right)}\right)} \\ $$

Question Number 146073 Answers: 0 Comments: 0

Let K be nonempty corps , K^∗ =K−{0_K } Prove that 1) Π_(x∈K^∗ ) x = −1 2)Deduce that p is prime ⇔ (p−1)!≡−1[p]

$${Let}\:{K}\:{be}\:{nonempty}\:\:{corps}\:,\:{K}^{\ast} ={K}−\left\{\mathrm{0}_{{K}} \right\} \\ $$$${Prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\underset{{x}\in{K}^{\ast} } {\prod}{x}\:=\:−\mathrm{1} \\ $$$$\left.\mathrm{2}\right){Deduce}\:{that}\: \\ $$$$\:\:{p}\:{is}\:{prime}\:\Leftrightarrow\:\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[{p}\right] \\ $$

Question Number 146067 Answers: 1 Comments: 0

transform the cartesian inyegral ∫_0 ^1 ∫_0 ^(√(1−x^2 )) e^(−(x^2 +y^2 )) dy dx into polar integral and evaluate it.

$${transform}\:{the}\:{cartesian}\:{inyegral}\: \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} {\int}}{e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} \:{dy}\:{dx}\:{into}\:{polar}\:{integral}\: \\ $$$${and}\:{evaluate}\:{it}. \\ $$

Pg 72 Pg 73 Pg 74 Pg 75 Pg 76 Pg 77 Pg 78 Pg 79 Pg 80 Pg 81

Terms of Service

Contact: info@tinkutara.com