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AllQuestion and Answers: Page 1786

Question Number 30855 Answers: 1 Comments: 0

∫((cosec^2 (x))/(√(cosecx+cotx)))dx

$$\int\frac{\mathrm{cosec}^{\mathrm{2}} \left(\mathrm{x}\right)}{\sqrt{\mathrm{cosecx}+\mathrm{cotx}}}\mathrm{dx} \\ $$

Question Number 30849 Answers: 0 Comments: 5

x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x+1=0 Σ_(k=1) ^7 [ℜ(x_k )]^2 = ? x_k = k^( th) root of the equation ℜ(x_k ) = real part of the root

$${x}^{\mathrm{7}} +{x}^{\mathrm{6}} +{x}^{\mathrm{5}} +{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0} \\ $$$$\: \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{7}} {\sum}}\left[\Re\left({x}_{{k}} \right)\right]^{\mathrm{2}} \:=\:? \\ $$$${x}_{{k}} \:=\:{k}^{\:\mathrm{th}} \:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\Re\left({x}_{{k}} \right)\:=\:\mathrm{real}\:\mathrm{part}\:\mathrm{of}\:\mathrm{the}\:\mathrm{root} \\ $$

Question Number 30840 Answers: 0 Comments: 1

Question Number 30836 Answers: 0 Comments: 1

Question Number 30827 Answers: 0 Comments: 0

Question Number 30823 Answers: 1 Comments: 0

Question Number 30820 Answers: 1 Comments: 0

Question Number 30813 Answers: 1 Comments: 0

Question Number 30808 Answers: 3 Comments: 0

Question Number 30803 Answers: 1 Comments: 0

oh how I wish this app has an AUDIO device system where we could explain all our doubts more accurately

$${oh}\:{how}\:{I}\:{wish}\:{this}\:{app}\:{has}\:{an} \\ $$$${AUDIO}\:{device}\:{system} \\ $$$${where}\:{we}\:{could}\:{explain} \\ $$$${all}\:{our}\:{doubts}\:{more} \\ $$$${accurately} \\ $$

Question Number 30798 Answers: 0 Comments: 1

find ∫_0 ^1 ((ln(1−x^2 ))/x^2 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 30796 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x^2 +(1/x^2 ))) dx.

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)} \:{dx}. \\ $$

Question Number 30787 Answers: 0 Comments: 0

Question Number 30783 Answers: 1 Comments: 0

Prove that determinant ((3,(a+b+c),(a^2 +b^2 +c^2 )),((a+b+c),(a^2 +b^2 +c^2 ),(a^3 +b^3 +c^3 )),((a^2 +b^2 +c^2 ),(a^3 +b^3 +c^3 ),(a^4 +b^4 +c^4 ))) =(a−b)^2 (b−c)^2 (c−a)^2

$${Prove}\:{that}\begin{vmatrix}{\mathrm{3}}&{{a}+{b}+{c}}&{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\\{{a}+{b}+{c}}&{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }&{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\\{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }&{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }&{{a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{c}^{\mathrm{4}} }\end{vmatrix} \\ $$$$=\left({a}−{b}\right)^{\mathrm{2}} \left({b}−{c}\right)^{\mathrm{2}} \left({c}−{a}\right)^{\mathrm{2}} \\ $$

Question Number 30780 Answers: 1 Comments: 0

Question Number 30777 Answers: 0 Comments: 0

find interms of n A_n = ∫_0 ^∞ ((ln(x))/((1+x^ )^n )) dx with n from N and n≥3 .

$${find}\:{interms}\:{of}\:{n}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}^{} \right)^{{n}} }\:{dx}\:{with}\:{n}\:{from} \\ $$$${N}\:{and}\:{n}\geqslant\mathrm{3}\:. \\ $$

Question Number 30776 Answers: 1 Comments: 1

find ∫_0 ^1 ((xdx)/((1+x^2 )(√(1−x^4 )))) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{xdx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:. \\ $$

Question Number 30775 Answers: 1 Comments: 1

letα ∈]0,π[ calculate ∫_0 ^(π/2) (dx/(2(cosα +chx))) .

$$\left.{let}\alpha\:\in\right]\mathrm{0},\pi\left[\:\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\mathrm{2}\left({cos}\alpha\:+{chx}\right)}\:.\right. \\ $$

Question Number 30774 Answers: 1 Comments: 1

find f(t)=∫_0 ^1 ln(1+tx^2 )dx with t>0

$${find}\:{f}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right){dx}\:\:{with}\:{t}>\mathrm{0} \\ $$

Question Number 30773 Answers: 0 Comments: 1

let a>0 calculate ∫_0 ^∞ (dx/((x^2 +a^2 )^2 )) 2) calculate ∫_(−∞) ^(+∞) (x^2 /((x^2 +a^2 )^2 ))dx.

$${let}\:{a}>\mathrm{0}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 30771 Answers: 0 Comments: 0

let I_n = ∫_0 ^(π/4) (dx/(cos^(2n+1) )) (n∈N) 1) find a and b fromR /∀x∈[0,(π/4)] (1/(cosx))=((acosx)/(1−sinx)) +((bcosx)/(1+sinx)) .find I_0 2) verify the relation (1/(cos^(2n+3) x))=(1/(cos^(2n+1) x)) +((sinx sinx)/(cos^(2n+3) )) .find the relation of recurrence between I_n and I_(n+1) .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dx}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} }\:\:\:\:\left({n}\in{N}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{and}\:{b}\:{fromR}\:/\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right] \\ $$$$\frac{\mathrm{1}}{{cosx}}=\frac{{acosx}}{\mathrm{1}−{sinx}}\:+\frac{{bcosx}}{\mathrm{1}+{sinx}}\:\:.{find}\:\:{I}_{\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:{verify}\:{the}\:{relation} \\ $$$$\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} {x}}=\frac{\mathrm{1}}{{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}\:+\frac{{sinx}\:{sinx}}{{cos}^{\mathrm{2}{n}+\mathrm{3}} }\:.{find}\:{the}\:{relation} \\ $$$${of}\:{recurrence}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}+\mathrm{1}} \:\:. \\ $$

Question Number 30769 Answers: 0 Comments: 1

find the value of I= ∫_0 ^1 (dx/((x+1)^2 (√(x^2 +2x +2)))) .

$${find}\:{the}\:{value}\:{of}\:{I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{2}}}\:. \\ $$

Question Number 30767 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ((1−(√x))/(1−^n (√x)))dx.

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−^{{n}} \sqrt{{x}}}{dx}. \\ $$

Question Number 30766 Answers: 0 Comments: 0

find ∫_0 ^1 (x/(√(x^4 +x^2 +1)))dx

$${find}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{x}}{\sqrt{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}}{dx} \\ $$

Question Number 30765 Answers: 0 Comments: 1

find ∫_0 ^1 (dx/(√(x^2 +x+1))) .

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}}\:. \\ $$

Question Number 30764 Answers: 0 Comments: 1

let I_n = ∫_0 ^(1/2) (1−2t)^n e^(−t) dt with n integr not 0 1) prove that ∀t∈[0,(1/2)] (1/(√e))(1−2t)^n ≤ (1−2t)^n e^(−t) ≤(1−2t)^n then find lim_(n→∞ ) I_n 2) prove that I_(n+1 ) =1−2(n+1)I_n 3) calculate I_1 ,I_2 , and I_3 .

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} {dt}\:\:{with}\:{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall{t}\in\left[\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$$$\frac{\mathrm{1}}{\sqrt{{e}}}\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \leqslant\:\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{e}^{−{t}} \:\leqslant\left(\mathrm{1}−\mathrm{2}{t}\right)^{{n}} \:{then}\:{find}\:{lim}_{{n}\rightarrow\infty\:} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{I}_{{n}+\mathrm{1}\:} =\mathrm{1}−\mathrm{2}\left({n}+\mathrm{1}\right){I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,\:{and}\:{I}_{\mathrm{3}} . \\ $$

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