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Question Number 96342 Answers: 2 Comments: 2

∫e^(sin(x)) dx

$$\int\mathfrak{e}^{\mathfrak{sin}\left(\mathfrak{x}\right)} \boldsymbol{\mathrm{d}}\mathfrak{x} \\ $$

Question Number 96340 Answers: 0 Comments: 4

The equations of two circles S_1 and S_2 are given by S_1 : x^2 + y^2 +2x +2y + 1 = 0 S_2 : x^2 + y^2 −4x + 2y +1 = 0. Show that S_1 and S_2 touch each other externally and obtain the equation of the common tangent T at the point of contact.

$$\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{two}\:\mathrm{circles}\:{S}_{\mathrm{1}} \:\mathrm{and}\:{S}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{given}\:\mathrm{by} \\ $$$$\:{S}_{\mathrm{1}} :\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{2}{y}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$$\:\:\:{S}_{\mathrm{2}} :\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:−\mathrm{4}{x}\:+\:\mathrm{2}{y}\:+\mathrm{1}\:=\:\mathrm{0}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:{S}_{\mathrm{1}} \:\mathrm{and}\:{S}_{\mathrm{2}} \:\mathrm{touch}\:\mathrm{each}\:\mathrm{other}\:\mathrm{externally}\:\mathrm{and}\:\mathrm{obtain} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{common}\:\mathrm{tangent}\:{T}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{contact}. \\ $$

Question Number 96335 Answers: 0 Comments: 0

∫(((3x^3 −x^2 +2x−4))/(√(x^3 −3x+4)))dx

$$\int\frac{\left(\mathrm{3}\mathfrak{x}^{\mathrm{3}} −\mathfrak{x}^{\mathrm{2}} +\mathrm{2}\mathfrak{x}−\mathrm{4}\right)}{\sqrt{\mathfrak{x}^{\mathrm{3}} −\mathrm{3}\mathfrak{x}+\mathrm{4}}}\boldsymbol{\mathrm{d}}\mathfrak{x} \\ $$

Question Number 96330 Answers: 0 Comments: 2

Question Number 96329 Answers: 0 Comments: 2

x⌊x⌊x⌊x⌋⌋⌋=88 x>0

$${x}\lfloor{x}\lfloor{x}\lfloor{x}\rfloor\rfloor\rfloor=\mathrm{88} \\ $$$${x}>\mathrm{0} \\ $$

Question Number 96321 Answers: 1 Comments: 1

It is given that x^2 =2^x . Find x.

$${It}\:{is}\:{given}\:{that}\:{x}^{\mathrm{2}} =\mathrm{2}^{{x}} .\:{Find}\:{x}. \\ $$

Question Number 96319 Answers: 1 Comments: 0

lim_(x→0) ((((1+x)^(1/x) )/e))^(1/x)

$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left(\frac{\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{{x}}} }{{e}}\right)^{\frac{\mathrm{1}}{{x}}} \\ $$

Question Number 96318 Answers: 0 Comments: 4

if (1+x)(1+x^2 ).....(1+x^(128) )=Σ_(r=0) ^n x^r then find n

$${if} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right).....\left(\mathrm{1}+{x}^{\mathrm{128}} \right)=\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}{x}^{{r}} \\ $$$${then}\:{find}\:{n} \\ $$

Question Number 96317 Answers: 0 Comments: 1

find the sum (1/4)+((1×3)/(4×6))+((1×3×5)/(4×6×8)).....=? find ∫((√x)/((√x)+(√(3−x))))dx

$${find}\:{the}\:{sum} \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}×\mathrm{3}}{\mathrm{4}×\mathrm{6}}+\frac{\mathrm{1}×\mathrm{3}×\mathrm{5}}{\mathrm{4}×\mathrm{6}×\mathrm{8}}.....=? \\ $$$${find} \\ $$$$\int\frac{\sqrt{{x}}}{\sqrt{{x}}+\sqrt{\mathrm{3}−{x}}}{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 96314 Answers: 0 Comments: 0

a\ Let E(x) denote the whole number part of the real number x, determine E(x^x ) and E(x^x^x ) for x∈]0,1[ b\ Calculate lim_(x→0) E(x^x^x )

$$\mathfrak{a}\backslash\:\mathcal{L}\mathfrak{et}\:\boldsymbol{\mathrm{E}}\left(\mathfrak{x}\right)\:\boldsymbol{\mathrm{d}}\mathfrak{enote}\:\mathfrak{the}\:\mathfrak{whole}\:\mathfrak{number}\:\mathfrak{part}\:\mathfrak{of}\:\mathfrak{the}\:\mathfrak{real} \\ $$$$\left.\mathfrak{number}\:\mathfrak{x},\:\boldsymbol{\mathrm{d}}\mathfrak{etermine}\:\boldsymbol{\mathrm{E}}\left(\mathfrak{x}^{\mathfrak{x}} \right)\:\mathfrak{an}\boldsymbol{\mathrm{d}}\:\boldsymbol{\mathrm{E}}\left(\mathfrak{x}^{\mathfrak{x}^{\mathfrak{x}} } \right)\:\mathfrak{for}\:\mathfrak{x}\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$$\mathfrak{b}\backslash\:\mathcal{C}\mathfrak{alculate}\:\underset{\mathfrak{x}\rightarrow\mathrm{0}} {\mathfrak{lim}}\boldsymbol{\mathrm{E}}\left(\mathfrak{x}^{\mathfrak{x}^{\mathfrak{x}} } \right) \\ $$

Question Number 96311 Answers: 1 Comments: 0

(4+(√(15)))^x + (4−(√(15)))^x = 62 x=?

$$\left(\mathrm{4}+\sqrt{\mathrm{15}}\right)^{{x}} \:+\:\left(\mathrm{4}−\sqrt{\mathrm{15}}\right)^{{x}} \:=\:\mathrm{62}\: \\ $$$${x}=? \\ $$

Question Number 96310 Answers: 0 Comments: 3

Question Number 96306 Answers: 0 Comments: 1

Given z = ((xy−4y^2 )/(x^2 +4y^2 )) , x,y≠0 find minimum and maximum value of z

$$\mathrm{Given}\:\mathrm{z}\:=\:\frac{\mathrm{xy}−\mathrm{4y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{4y}^{\mathrm{2}} }\:,\:\mathrm{x},\mathrm{y}\neq\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{z}\: \\ $$

Question Number 96302 Answers: 0 Comments: 0

find ∫_0 ^∞ ((arctan(2x))/(1+x^2 ))dx

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

Question Number 96296 Answers: 1 Comments: 0

Question Number 96293 Answers: 1 Comments: 0

Question Number 96290 Answers: 0 Comments: 1

If : tan(x +iy) = a + bi then find a,b

$${If}\::\:\mathrm{tan}\left({x}\:+{iy}\right)\:=\:{a}\:+\:{bi}\: \\ $$$${then}\:{find}\:{a},{b} \\ $$

Question Number 96289 Answers: 0 Comments: 1

1010^x +2020^x =4040^x x=?

$$\mathrm{1010}^{{x}} +\mathrm{2020}^{{x}} =\mathrm{4040}^{{x}} \\ $$$${x}=? \\ $$

Question Number 96287 Answers: 1 Comments: 0

Consider the system in N^3 (S): { ((p^2 +q^2 =r^2 )),((q+p+r=24)),((r<p+q)) :} Show that the triplet (p:q:r) is solution to (S) if and only if r<12. p and q are solutions to the equation; n^2 −(24−r)n+24(12−r)=0 where n is an unknown.p

$$\mathcal{C}\mathrm{onsider}\:\mathrm{the}\:\mathrm{system}\:\mathrm{in}\:\mathbb{N}^{\mathrm{3}} \\ $$$$\left(\mathrm{S}\right):\:\begin{cases}{\mathrm{p}^{\mathrm{2}} +\mathrm{q}^{\mathrm{2}} =\mathrm{r}^{\mathrm{2}} }\\{\mathrm{q}+\mathrm{p}+\mathrm{r}=\mathrm{24}}\\{\mathrm{r}<\mathrm{p}+\mathrm{q}}\end{cases} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{triplet}\:\left(\mathrm{p}:\mathrm{q}:\mathrm{r}\right)\:\mathrm{is}\:\mathrm{solution}\:\mathrm{to}\:\left(\mathrm{S}\right)\:\mathrm{if} \\ $$$$\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:\mathrm{r}<\mathrm{12}.\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{are}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}; \\ $$$$\mathrm{n}^{\mathrm{2}} −\left(\mathrm{24}−\mathrm{r}\right)\mathrm{n}+\mathrm{24}\left(\mathrm{12}−\mathrm{r}\right)=\mathrm{0}\:\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{an}\:\mathrm{unknown}.\mathrm{p} \\ $$

Question Number 96282 Answers: 3 Comments: 0