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Question Number 181749 Answers: 1 Comments: 0

Question Number 181879 Answers: 0 Comments: 0

∫ ((arccos(tg^3 (2x)))/(1+4x^2 )) dx

$$\:\:\:\int\:\:\frac{\boldsymbol{\mathrm{arccos}}\left(\boldsymbol{\mathrm{tg}}^{\mathrm{3}} \left(\mathrm{2}\boldsymbol{\mathrm{x}}\right)\right)}{\mathrm{1}+\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 181699 Answers: 0 Comments: 1

Prove it by mathematical induction: ∣ Σ_(j=1) ^n x_j ∣ ≤ Σ_(j=1) ^n sin x_j ; x_j ∈ [ 0 , π ]

$$\mathrm{Prove}\:\mathrm{it}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}: \\ $$$$\mid\:\:\underset{\boldsymbol{\mathrm{j}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{x}_{\boldsymbol{\mathrm{j}}} \:\:\mid\:\:\leqslant\:\:\underset{\boldsymbol{\mathrm{j}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{sin}\:\mathrm{x}_{\boldsymbol{\mathrm{j}}} \:\:\:\:\:;\:\:\:\:\:\mathrm{x}_{\boldsymbol{\mathrm{j}}} \:\in\:\left[\:\mathrm{0}\:,\:\pi\:\right] \\ $$

Question Number 181685 Answers: 0 Comments: 0

determinant (((ty jik),(gf ),(cf),(ior)),(,,,),(,,,))

$$\begin{array}{|c|c|c|}{\mathrm{ty}\:\mathrm{jik}}&\hline{\mathrm{gf}\:}&\hline{\mathrm{cf}}&\hline{\mathrm{ior}}\\{}&\hline{}&\hline{}&\hline{}\\{}&\hline{}&\hline{}&\hline{}\\\hline\end{array} \\ $$

Question Number 181681 Answers: 0 Comments: 0

Question Number 181679 Answers: 0 Comments: 0

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Question Number 181677 Answers: 0 Comments: 0

the distirbution function for a random variable X is F(x)= { ((1−2e^(−2x) x≥0)),((0 x<0)) :} find 1)the density function f(x) 2)p(x>2) 3)p(−3<x≤4)

$$\:{the}\:{distirbution}\:{function}\:{for}\:{a}\:{random} \\ $$$${variable}\:{X}\:{is} \\ $$$${F}\left({x}\right)=\begin{cases}{\mathrm{1}−\mathrm{2}{e}^{−\mathrm{2}{x}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\geqslant\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}<\mathrm{0}}\end{cases} \\ $$$${find} \\ $$$$\left.\mathrm{1}\right){the}\:{density}\:{function}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){p}\left({x}>\mathrm{2}\right) \\ $$$$\left.\mathrm{3}\right){p}\left(−\mathrm{3}<{x}\leqslant\mathrm{4}\right) \\ $$

Question Number 181676 Answers: 4 Comments: 0

Question Number 181678 Answers: 0 Comments: 0

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Question Number 182202 Answers: 0 Comments: 0

Question Number 181673 Answers: 4 Comments: 0

Question Number 181658 Answers: 0 Comments: 4

Area of Circle ?

$${Area}\:{of}\:{Circle}\:? \\ $$

Question Number 181651 Answers: 4 Comments: 7

Question Number 181644 Answers: 2 Comments: 0

prove that lim_(x→∞) (((1+(1/3)+(1/5)+∙∙∙∙+(1/(2x+1)))/(xln(√x))))^(ln(√x)) =2e^(γ/2)

$${prove}\:{that} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}+\centerdot\centerdot\centerdot\centerdot+\frac{\mathrm{1}}{\mathrm{2}{x}+\mathrm{1}}}{{xln}\sqrt{{x}}}\right)^{{ln}\sqrt{{x}}} =\mathrm{2}{e}^{\frac{\gamma}{\mathrm{2}}} \: \\ $$$$ \\ $$

Question Number 181643 Answers: 0 Comments: 2

Ω=∫_0 ^(π/2) tan^(−1) cos x dx (I′d need the exact value if possible. I′ve got no idea if and how this can be solved.)

$$\Omega=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\mathrm{tan}^{−\mathrm{1}} \:\mathrm{cos}\:{x}\:{dx} \\ $$$$\left(\mathrm{I}'\mathrm{d}\:\mathrm{need}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{value}\:\mathrm{if}\:\mathrm{possible}.\:\mathrm{I}'\mathrm{ve}\right. \\ $$$$\left.\mathrm{got}\:\mathrm{no}\:\mathrm{idea}\:\mathrm{if}\:\mathrm{and}\:\mathrm{how}\:\mathrm{this}\:\mathrm{can}\:\mathrm{be}\:\mathrm{solved}.\right) \\ $$

Question Number 181628 Answers: 1 Comments: 0

find the domain and range of y = (1/((x − 1)(x + 2))) restricted to 0 ≤ x ≤ 6

$$\mathrm{find}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{and}\:\mathrm{range}\:\mathrm{of}\:\:\:\mathrm{y}\:\:=\:\:\frac{\mathrm{1}}{\left(\mathrm{x}\:\:−\:\:\mathrm{1}\right)\left(\mathrm{x}\:\:+\:\:\mathrm{2}\right)} \\ $$$$\mathrm{restricted}\:\mathrm{to}\:\:\:\mathrm{0}\:\:\leqslant\:\:\mathrm{x}\:\:\leqslant\:\:\mathrm{6} \\ $$

Question Number 181625 Answers: 0 Comments: 0

K-Lemoine′s , I-incenter in △ABC. Prove that: KA^4 +KB^4 +KC^4 ≥ IA^4 +IB^4 +IC^4

$$\mathrm{K}-\mathrm{Lemoine}'\mathrm{s}\:,\:\mathrm{I}-\mathrm{incenter}\:\mathrm{in}\:\bigtriangleup\mathrm{ABC}. \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{KA}^{\mathrm{4}} +\mathrm{KB}^{\mathrm{4}} +\mathrm{KC}^{\mathrm{4}} \:\geqslant\:\mathrm{IA}^{\mathrm{4}} +\mathrm{IB}^{\mathrm{4}} +\mathrm{IC}^{\mathrm{4}} \\ $$

Question Number 181627 Answers: 1 Comments: 1

Point E is marked on side AD in rhombus ABCD. If AC = 6 (√(10)) and BD = 2 (√(10)) . How many different integer values can a piece of BE take?

$$\mathrm{Point}\:\:\mathrm{E}\:\:\mathrm{is}\:\mathrm{marked}\:\mathrm{on}\:\mathrm{side}\:\:\mathrm{AD}\:\:\mathrm{in} \\ $$$$\mathrm{rhombus}\:\:\mathrm{ABCD}.\:\mathrm{If}\:\:\mathrm{AC}\:=\:\mathrm{6}\:\sqrt{\mathrm{10}}\:\:\mathrm{and} \\ $$$$\mathrm{BD}\:=\:\mathrm{2}\:\sqrt{\mathrm{10}}\:.\:\mathrm{How}\:\mathrm{many}\:\mathrm{different} \\ $$$$\mathrm{integer}\:\mathrm{values}\:\mathrm{can}\:\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\:\mathrm{BE}\:\:\mathrm{take}? \\ $$

Question Number 181619 Answers: 2 Comments: 1

Question Number 181618 Answers: 2 Comments: 0

(dy/dx)+2xy=x^2 y(0)=3 Solve .

$$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{2xy}=\mathrm{x}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{y}\left(\mathrm{0}\right)=\mathrm{3} \\ $$$$ \\ $$$$\mathrm{Solve} \\ $$$$ \\ $$$$. \\ $$

Question Number 181617 Answers: 0 Comments: 0

Solve: x(dy/dx)+(x+1)y=e^x^2 .

$$\mathrm{Solve}: \\ $$$$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\mathrm{x}+\mathrm{1}\right)\mathrm{y}=\mathrm{e}^{\mathrm{x}^{\mathrm{2}} } \\ $$$$ \\ $$$$. \\ $$

Question Number 181615 Answers: 1 Comments: 0

Determine whether this is Homogenous or not (dy/dx)=(y/(y−2x)) .

$$\mathrm{Determine}\:\mathrm{whether}\:\mathrm{this}\:\mathrm{is}\:\mathrm{Homogenous} \\ $$$$\mathrm{or}\:\mathrm{not} \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{y}−\mathrm{2x}} \\ $$$$ \\ $$$$. \\ $$

Question Number 181613 Answers: 1 Comments: 0

(dy/dx)=((xy+y^2 )/x^2 ) y(−1)=2 .

$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{xy}+\mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{y}\left(−\mathrm{1}\right)=\mathrm{2} \\ $$$$ \\ $$$$. \\ $$

Question Number 181607 Answers: 0 Comments: 0

Question Number 181605 Answers: 0 Comments: 1

Question Number 181599 Answers: 1 Comments: 0

f(x) is a strictly monotonic function in its domain (0, +∞) such that ∀x>0, f(f(x)−(1/x))=2. Find f(x).

$${f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{strictly}\:\mathrm{monotonic}\:\mathrm{function}\:\mathrm{in}\:\mathrm{its}\:\mathrm{domain}\:\left(\mathrm{0},\:+\infty\right) \\ $$$$\mathrm{such}\:\mathrm{that}\:\forall{x}>\mathrm{0},\:{f}\left({f}\left({x}\right)−\frac{\mathrm{1}}{{x}}\right)=\mathrm{2}. \\ $$$$\mathrm{Find}\:{f}\left({x}\right). \\ $$

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