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Question Number 116272    Answers: 1   Comments: 0

∫_0 ^(π/2) ln(x^2 +ln^2 (cos(x)))dx=πln(ln(2)) posted Quation not solved yet i hop someon Giv idea for this one thank you

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({x}^{\mathrm{2}} +{ln}^{\mathrm{2}} \left({cos}\left({x}\right)\right)\right){dx}=\pi{ln}\left({ln}\left(\mathrm{2}\right)\right) \\ $$$${posted}\:{Quation}\: \\ $$$${not}\:{solved}\:{yet}\:{i}\:{hop}\:{someon}\:{Giv}\:{idea}\:{for} \\ $$$${this}\:{one}\:{thank}\:{you} \\ $$

Question Number 116271    Answers: 2   Comments: 0

∫_(−1) ^( 0) [ln(ln(1+x))]^2 dx

$$\int_{−\mathrm{1}} ^{\:\mathrm{0}} \left[{ln}\left({ln}\left(\mathrm{1}+{x}\right)\right)\right]^{\mathrm{2}} {dx} \\ $$

Question Number 116267    Answers: 2   Comments: 0

Find the equation of parabola that passes through points (1,2) ,(3,4) and tangents to the line y=−x+((25)/3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{parabola}\:\mathrm{that} \\ $$$$\mathrm{passes}\:\mathrm{through}\:\mathrm{points}\:\left(\mathrm{1},\mathrm{2}\right)\:,\left(\mathrm{3},\mathrm{4}\right)\: \\ $$$$\mathrm{and}\:\mathrm{tangents}\:\mathrm{to}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=−\mathrm{x}+\frac{\mathrm{25}}{\mathrm{3}} \\ $$

Question Number 116259    Answers: 1   Comments: 0

Question Number 116253    Answers: 0   Comments: 0

(1) Show that Σ_(i = 0) ^n L_i (x) = 1 (2) Show that Σ_(i = 0) ^n L_i (x). x_i ^k = x^k , k ≤ n

$$\left(\mathrm{1}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right)\:\:\:=\:\:\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\:\:\:\underset{\mathrm{i}\:\:=\:\:\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\:\mathrm{L}_{\mathrm{i}} \left(\mathrm{x}\right).\:\mathrm{x}_{\mathrm{i}} ^{\mathrm{k}} \:\:\:=\:\:\:\mathrm{x}^{\mathrm{k}} ,\:\:\:\:\:\:\:\:\mathrm{k}\:\leqslant\:\mathrm{n} \\ $$

Question Number 116252    Answers: 2   Comments: 0

lim_(x→−∞) ((e^x +e^(−x) )/(e^x −e^(−x) )) = ?

$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} }{\mathrm{e}^{\mathrm{x}} −\mathrm{e}^{−\mathrm{x}} }\:=\:? \\ $$

Question Number 116250    Answers: 0   Comments: 0

1)explicite U_n =∫_0 ^∞ e^(−n[x]) cos(3[x])dx 2) calculate lim_(n→+∞) U_n 3)find nsture of Σ U_n

$$\left.\mathrm{1}\right)\mathrm{explicite}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{n}\left[\mathrm{x}\right]} \mathrm{cos}\left(\mathrm{3}\left[\mathrm{x}\right]\right)\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{U}_{\mathrm{n}} \\ $$$$\left.\mathrm{3}\right)\mathrm{find}\:\mathrm{nsture}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \\ $$

Question Number 116249    Answers: 0   Comments: 0

find ∫_0 ^1 ((arctan(x^2 +3))/(x^2 +3))dx

$$\mathrm{find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{\mathrm{arctan}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{3}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{3}}\mathrm{dx} \\ $$

Question Number 116248    Answers: 0   Comments: 0

calculate ∫_(−∞) ^∞ ((arctan(2+x^2 ))/(x^2 −x +1))dx

$$\mathrm{calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{2}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} −\mathrm{x}\:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 116247    Answers: 0   Comments: 0

find the value of I =∫_0 ^∞ ((ch(cos(2x)))/(x^2 +9))dx and J =∫_0 ^∞ ((cos(ch(2x)))/(x^2 +9))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{ch}\left(\mathrm{cos}\left(\mathrm{2x}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{9}}\mathrm{dx}\:\mathrm{and} \\ $$$$\mathrm{J}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{cos}\left(\mathrm{ch}\left(\mathrm{2x}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{9}}\mathrm{dx} \\ $$

Question Number 116245    Answers: 3   Comments: 0

calculate ∫_1 ^∞ (dx/((2x^2 −1)^5 ))

$$\mathrm{calculate}\:\:\int_{\mathrm{1}} ^{\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{2x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{5}} } \\ $$

Question Number 116239    Answers: 4   Comments: 0

a circle is tangent to x−axis , y−axis and the line 3x−4y+6=0. what its the equation?

$$\mathrm{a}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{x}−\mathrm{axis}\:,\:\mathrm{y}−\mathrm{axis} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{line}\:\mathrm{3x}−\mathrm{4y}+\mathrm{6}=\mathrm{0}. \\ $$$$\mathrm{what}\:\mathrm{its}\:\mathrm{the}\:\mathrm{equation}? \\ $$

Question Number 116238    Answers: 1   Comments: 0

How long does it take for the disintegration of (3/(10)) of an atom with radioactive constant λ ?

$$\mathrm{How}\:\mathrm{long}\:\mathrm{does}\:\mathrm{it}\:\mathrm{take}\:\mathrm{for}\:\mathrm{the}\:\mathrm{disintegration}\:\mathrm{of} \\ $$$$\frac{\mathrm{3}}{\mathrm{10}}\:\mathrm{of}\:\mathrm{an}\:\mathrm{atom}\:\mathrm{with}\:\mathrm{radioactive}\:\mathrm{constant}\:\lambda\:? \\ $$

Question Number 116237    Answers: 2   Comments: 0

∫ (√(5cos^2 x+4)) dx ?

$$\int\:\sqrt{\mathrm{5cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{4}}\:\mathrm{dx}\:? \\ $$

Question Number 116231    Answers: 1   Comments: 0

∫ sec x tan x (√(tan^2 x−3)) dx ?

$$\int\:\mathrm{sec}\:\mathrm{x}\:\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}−\mathrm{3}}\:\mathrm{dx}\:? \\ $$

Question Number 116221    Answers: 3   Comments: 0

Given α,β and ϕ are the roots of x^3 −px^2 +qx−pq = 0 . Find the value of (α/β)+(β/α)+(β/ϕ)+(ϕ/β)+(α/ϕ)+(ϕ/α)=?

$$\mathrm{Given}\:\alpha,\beta\:\mathrm{and}\:\varphi\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\mathrm{x}^{\mathrm{3}} −\mathrm{px}^{\mathrm{2}} +\mathrm{qx}−\mathrm{pq}\:=\:\mathrm{0}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\varphi}+\frac{\varphi}{\beta}+\frac{\alpha}{\varphi}+\frac{\varphi}{\alpha}=? \\ $$

Question Number 116216    Answers: 1   Comments: 0

∫_0 ^π ((ln (1+(1/2)cos x))/(cos x)) dx ?

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$

Question Number 116192    Answers: 2   Comments: 1

Question Number 116184    Answers: 1   Comments: 0

How many 6-digits positive integers which are formed by the digits 1 to 9 are such that each of the digits in the number appears at least twice? [For instance: 121233,122221,777777 and etc.]

$$\mathrm{How}\:\mathrm{many}\:\mathrm{6}-\mathrm{digits}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{formed}\:\mathrm{by}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{1}\:\mathrm{to}\:\mathrm{9}\:\mathrm{are}\:\mathrm{such}\:\mathrm{that}\:\mathrm{each} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{the}\:\mathrm{number}\:\mathrm{appears}\:\mathrm{at}\:\mathrm{least} \\ $$$$\mathrm{twice}?\:\left[\mathrm{For}\:\mathrm{instance}:\:\mathrm{121233},\mathrm{122221},\mathrm{777777}\:\mathrm{and}\:\mathrm{etc}.\right] \\ $$

Question Number 116177    Answers: 0   Comments: 1

Question Number 116173    Answers: 0   Comments: 2

How many ways can the letters of the word DENKENMATHEMATICAL be arranged if no same letters must be together

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\:\mathrm{DENKENMATHEMATICAL} \\ $$$$\mathrm{be}\:\mathrm{arranged}\:\mathrm{if}\:\mathrm{no}\:\mathrm{same}\:\mathrm{letters}\:\mathrm{must}\:\mathrm{be}\:\mathrm{together} \\ $$

Question Number 116167    Answers: 1   Comments: 0

Show that ∀n≥2 the equation x^n =x+n admits a unique solution u_n ∈(1,2]

$$\mathrm{Show}\:\mathrm{that}\:\forall{n}\geqslant\mathrm{2}\:\mathrm{the}\:\mathrm{equation}\:{x}^{{n}} ={x}+{n} \\ $$$$\mathrm{admits}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{u}_{\mathrm{n}} \in\left(\mathrm{1},\mathrm{2}\right] \\ $$

Question Number 116166    Answers: 1   Comments: 0

∀n∈N^∗ , suppose u_n =(5sin(1/n^2 )+(1/5)cos n)^n Prove that lim_(n→+∞) u_n =0

$$\forall{n}\in\mathbb{N}^{\ast} ,\:\mathrm{suppose}\:{u}_{{n}} =\left(\mathrm{5sin}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}}\mathrm{cos}\:{n}\right)^{{n}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{n}\rightarrow+\infty} {\mathrm{lim}}{u}_{{n}} =\mathrm{0} \\ $$

Question Number 116162    Answers: 1   Comments: 0

Σ_(n=0) ^∞ ((2n+1)/(16^n (n^2 +3n+2))) (((2n)),(n) )^2 =(8/(3π)) m.n.july 1970.

$$\:\: \\ $$$$ \\ $$$$\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{16}^{{n}} \left({n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}\right)}\:\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix}^{\mathrm{2}} \:=\frac{\mathrm{8}}{\mathrm{3}\pi}\: \\ $$$$ \\ $$$${m}.{n}.{july}\:\mathrm{1970}. \\ $$$$\: \\ $$

Question Number 116163    Answers: 1   Comments: 0

I need a general rule for factorising 1) a^n +b^n 2) a^n −b^n when n is even and when n is odd.

$$\mathrm{I}\:\mathrm{need}\:\mathrm{a}\:\mathrm{general}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{factorising} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{a}^{\mathrm{n}} +\mathrm{b}^{\mathrm{n}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{a}^{\mathrm{n}} −\mathrm{b}^{\mathrm{n}} \\ $$$$\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{when}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}. \\ $$

Question Number 116196    Answers: 6   Comments: 0

please solve : ∫_0 ^( (π/4)) tan^9 (x)dx =???

$$\:\:\:\:\:{please}\:{solve}\:: \\ $$$$\:\:\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {tan}^{\mathrm{9}} \left({x}\right){dx}\:=??? \\ $$

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