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Question Number 110197    Answers: 2   Comments: 4

(−1)^π =? (−1)^((22)/7) =(−1)^(3+(1/7)) =(−1)^3 .(−1)^(1/7) =−(−1)^(1/7) let (−1)^π =t ⇒−(−1)^(1/7) =t ⇒(−1)^(1/7) =−t ⇒(−1)=(−t)^7 ⇒−1=−t^7 ⇒1=t^7 hence t=1 ∴(−1)^π =1 i request all math professionals to check this and if any error then pls comment.

$$\left(−\mathrm{1}\right)^{\pi} =? \\ $$$$\left(−\mathrm{1}\right)^{\frac{\mathrm{22}}{\mathrm{7}}} \\ $$$$=\left(−\mathrm{1}\right)^{\mathrm{3}+\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$$=\left(−\mathrm{1}\right)^{\mathrm{3}} .\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$$=−\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$${let}\:\left(−\mathrm{1}\right)^{\pi} ={t} \\ $$$$\Rightarrow−\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}} ={t} \\ $$$$\Rightarrow\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}} =−{t} \\ $$$$\Rightarrow\left(−\mathrm{1}\right)=\left(−{t}\right)^{\mathrm{7}} \\ $$$$\Rightarrow−\mathrm{1}=−{t}^{\mathrm{7}} \\ $$$$\Rightarrow\mathrm{1}={t}^{\mathrm{7}} \\ $$$${hence}\:{t}=\mathrm{1} \\ $$$$\therefore\left(−\mathrm{1}\right)^{\pi} =\mathrm{1} \\ $$$$\boldsymbol{{i}}\:\boldsymbol{{request}}\:\boldsymbol{{all}}\:\boldsymbol{{math}} \\ $$$$\boldsymbol{{professionals}}\:\boldsymbol{{to}}\: \\ $$$$\boldsymbol{{check}}\:\boldsymbol{{this}}\:\boldsymbol{{and}}\:\boldsymbol{{if}} \\ $$$$\boldsymbol{{any}}\:\boldsymbol{{error}}\:\boldsymbol{{then}}\:\boldsymbol{{pls}} \\ $$$$\boldsymbol{{comment}}. \\ $$

Question Number 110183    Answers: 3   Comments: 0

(√★)((be)/(math))(√★) log _2 (x)+log _3 (x)+log _4 (x)=1 x=?

$$\:\:\:\sqrt{\bigstar}\frac{{be}}{{math}}\sqrt{\bigstar} \\ $$$$\:\:\mathrm{log}\:_{\mathrm{2}} \left({x}\right)+\mathrm{log}\:_{\mathrm{3}} \left({x}\right)+\mathrm{log}\:_{\mathrm{4}} \left({x}\right)=\mathrm{1} \\ $$$$\:\:\:{x}=? \\ $$

Question Number 110182    Answers: 1   Comments: 2

Solve x^3 +15x−92=0

$$\mathrm{Solve}\:{x}^{\mathrm{3}} +\mathrm{15}{x}−\mathrm{92}=\mathrm{0} \\ $$

Question Number 110175    Answers: 2   Comments: 0

solve the integral ∫_(−∞) ^(+∞) ((sinx)/(x)^(1/3) )dx

$${solve}\:{the}\:{integral} \\ $$$$\int_{−\infty} ^{+\infty} \frac{\mathrm{sin}{x}}{\sqrt[{\mathrm{3}}]{{x}}}{dx} \\ $$

Question Number 110173    Answers: 2   Comments: 0

If Σ_(r=1) ^n t_r =((n(n+1)(n+2)(n+3))/8) then lim_(n→∞) Σ_(r=1) ^n (1/t_r ) = ?

$${If}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{t}_{{r}} =\frac{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)}{\mathrm{8}} \\ $$$${then}\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{t}_{{r}} }\:=\:? \\ $$$$ \\ $$

Question Number 110157    Answers: 2   Comments: 0

If we have 5 people, how many ways can they be seated on a round table, if there are, (a) 7 chairs (b) 3 chairs available

$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\:\:\mathrm{5}\:\:\mathrm{people},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{they}\:\mathrm{be}\:\mathrm{seated} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{round}\:\mathrm{table},\:\mathrm{if}\:\mathrm{there}\:\mathrm{are}, \\ $$$$\left(\mathrm{a}\right)\:\:\:\mathrm{7}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\mathrm{3}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\:\:\:\:\mathrm{available} \\ $$

Question Number 110156    Answers: 1   Comments: 0

If we have 5 people, how many ways can they be seated in a row on a chair, if their are, (a) 7 chairs (b) 3 chairs available

$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\:\:\mathrm{5}\:\mathrm{people},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{they}\:\mathrm{be}\:\mathrm{seated}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{chair},\:\mathrm{if}\:\mathrm{their}\:\mathrm{are}, \\ $$$$\left(\mathrm{a}\right)\:\:\:\mathrm{7}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\mathrm{3}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\:\mathrm{available} \\ $$

Question Number 110154    Answers: 0   Comments: 0

prove that ∫_0 ^1 ∫_0 ^1 ((tanh^(−1) (^4 (√x))tanh^(−1) (^4 (√y)))/(x(√y)))=π^2

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{tanh}^{−\mathrm{1}} \left(^{\mathrm{4}} \sqrt{{x}}\right)\mathrm{tanh}^{−\mathrm{1}} \left(^{\mathrm{4}} \sqrt{{y}}\right)}{{x}\sqrt{{y}}}=\pi^{\mathrm{2}} \\ $$

Question Number 110149    Answers: 0   Comments: 1

find the domain f(x,y)=x+4(√(y )) ?

$${find}\:{the}\:{domain}\:{f}\left({x},{y}\right)={x}+\mathrm{4}\sqrt{{y}\:}\:? \\ $$

Question Number 110145    Answers: 2   Comments: 0

prove that ∫_1 ^∞ ((ln(x))/(1+x+x^2 +x^3 ))dx=(G/2)−(π^2 /(32)) G(catalan constant)

$${prove}\:{that} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{ln}\left({x}\right)}{\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} }{dx}=\frac{{G}}{\mathrm{2}}−\frac{\pi^{\mathrm{2}} }{\mathrm{32}} \\ $$$${G}\left({catalan}\:{constant}\right) \\ $$

Question Number 110143    Answers: 2   Comments: 0

Question Number 110136    Answers: 4   Comments: 0

Question Number 110132    Answers: 0   Comments: 4

lim_(x→0) ((5cos^2 x−2cos x−3)/(cos x−cos 3x)) ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{5cos}\:^{\mathrm{2}} {x}−\mathrm{2cos}\:{x}−\mathrm{3}}{\mathrm{cos}\:{x}−\mathrm{cos}\:\mathrm{3}{x}}\:? \\ $$

Question Number 110118    Answers: 1   Comments: 4

prove that ∫_0 ^1 Γ(1−(x/2))Γ(1+(x/2))dx=(4/π)G where G(catalan constant)

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \Gamma\left(\mathrm{1}−\frac{{x}}{\mathrm{2}}\right)\Gamma\left(\mathrm{1}+\frac{{x}}{\mathrm{2}}\right){dx}=\frac{\mathrm{4}}{\pi}{G} \\ $$$${where}\:{G}\left({catalan}\:{constant}\right) \\ $$

Question Number 110119    Answers: 1   Comments: 0

A body of mass 5kg is acted on by two forces 30N and 40N. Find the vector acceleration of the object?

$$\mathrm{A}\:\mathrm{body}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{5kg}\:\mathrm{is}\:\mathrm{acted}\:\mathrm{on}\:\mathrm{by}\:\mathrm{two}\:\mathrm{forces}\:\mathrm{30N} \\ $$$$\mathrm{and}\:\mathrm{40N}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{vector}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{the}\:\mathrm{object}? \\ $$

Question Number 110112    Answers: 1   Comments: 0

show that ∫_0 ^∞ xsin(x^3 )dx=(1/3)•(π/(Γ((1/3))))

$${show}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}\mathrm{sin}\left({x}^{\mathrm{3}} \right){dx}=\frac{\mathrm{1}}{\mathrm{3}}\bullet\frac{\pi}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$

Question Number 110109    Answers: 2   Comments: 0

Question Number 110096    Answers: 0   Comments: 8

find the following product integral (1) ∫(x)^dx (2) ∫(e^x )^dx

$${find}\:{the}\:{following}\:{product}\:{integral} \\ $$$$\:\:\:\:\:\:\:\:\left(\mathrm{1}\right)\:\:\:\:\int\left({x}\right)^{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)\:\:\:\:\int\left({e}^{{x}} \right)^{{dx}} \\ $$

Question Number 110095    Answers: 3   Comments: 0

[((be)/(math))] lim_(x→0) x^2 cos ((1/x))

$$\:\:\:\left[\frac{{be}}{{math}}\right] \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \:\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right) \\ $$

Question Number 110092    Answers: 1   Comments: 0

Question Number 110087    Answers: 0   Comments: 3

Solve the system following of equations { ((x+y+z=2)),((2x+3y+z=1)),((x^2 +(y+2)^2 +(z−1)^2 =9)) :}

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{following}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\begin{cases}{\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{2}}\\{\mathrm{2x}+\mathrm{3y}+\mathrm{z}=\mathrm{1}}\\{\mathrm{x}^{\mathrm{2}} +\left(\mathrm{y}+\mathrm{2}\right)^{\mathrm{2}} +\left(\mathrm{z}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{9}}\end{cases} \\ $$

Question Number 110086    Answers: 1   Comments: 0

((♠JS♠)/(★■.★)) If lim_(x→a) ((x^2 +2∣ax∣−3a^2 )/( (√x)−(√a) )) = P , with a>0 then the value of lim_(x→a) ((2x^2 −∣ax∣−a^2 )/(x−a)) is ___ (a) ((3P)/(4(√a))) (b) ((3P)/(8(√a))) (c) ((8P)/(3(√a))) (d) ((4P)/(3(√a) )) (e) ((3P)/(8a))

$$\:\:\frac{\spadesuit{JS}\spadesuit}{\bigstar\blacksquare.\bigstar} \\ $$$${If}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} +\mathrm{2}\mid{ax}\mid−\mathrm{3}{a}^{\mathrm{2}} }{\:\sqrt{{x}}−\sqrt{{a}}\:}\:=\:{P}\:,\:{with}\:{a}>\mathrm{0} \\ $$$${then}\:{the}\:{value}\:{of}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{\mathrm{2}{x}^{\mathrm{2}} −\mid{ax}\mid−{a}^{\mathrm{2}} }{{x}−{a}}\:{is} \\ $$$$\_\_\_ \\ $$$$\:\left({a}\right)\:\frac{\mathrm{3}{P}}{\mathrm{4}\sqrt{{a}}}\:\:\:\:\:\:\left({b}\right)\:\frac{\mathrm{3}{P}}{\mathrm{8}\sqrt{{a}}}\:\:\:\:\:\left({c}\right)\:\frac{\mathrm{8}{P}}{\mathrm{3}\sqrt{{a}}} \\ $$$$\:\:\left({d}\right)\:\frac{\mathrm{4}{P}}{\mathrm{3}\sqrt{{a}}\:}\:\:\:\:\left({e}\right)\:\frac{\mathrm{3}{P}}{\mathrm{8}{a}} \\ $$

Question Number 110056    Answers: 2   Comments: 0

prove that ∫_(−∞) ^(+∞) ((cos^4 x−6sin^2 xcos^2 x+sin^4 x)/(1+x^2 ))dx=(π/e^4 )

$${prove}\:{that}\: \\ $$$$\int_{−\infty} ^{+\infty} \frac{\mathrm{cos}^{\mathrm{4}} {x}−\mathrm{6sin}^{\mathrm{2}} {x}\mathrm{cos}^{\mathrm{2}} {x}+\mathrm{sin}^{\mathrm{4}} {x}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=\frac{\pi}{{e}^{\mathrm{4}} } \\ $$

Question Number 110050    Answers: 1   Comments: 0

Question Number 110049    Answers: 4   Comments: 0

Question Number 110045    Answers: 0   Comments: 0

let x ∈ ]0,1[ find E(x^x ) and E(x^x^x )

$$\left.{let}\:{x}\:\in\:\right]\mathrm{0},\mathrm{1}\left[\:\:\right. \\ $$$${find}\:{E}\left({x}^{{x}} \right)\:{and}\:{E}\left({x}^{{x}^{{x}} } \right) \\ $$

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