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Question Number 71311 Answers: 1 Comments: 1

Question Number 71301 Answers: 0 Comments: 0

let f(x)=∫_(1+x) ^(1+x^2 ) ((arctan(xt+2))/(x+t))dt calculate f^′ (x) 2)find lim_(x→0) f(x)

$${let}\:{f}\left({x}\right)=\int_{\mathrm{1}+{x}} ^{\mathrm{1}+{x}^{\mathrm{2}} } \:\frac{{arctan}\left({xt}+\mathrm{2}\right)}{{x}+{t}}{dt} \\ $$$${calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right) \\ $$

Question Number 71362 Answers: 0 Comments: 0

please prove that lim_(x→0) ((x−sinx)/x^3 ) =(1/6) by using x=3y and sin3y=3siny−4sin^3 y

$$\boldsymbol{\mathrm{please}}\:\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\frac{\boldsymbol{{x}}−\boldsymbol{{sinx}}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\frac{\mathrm{1}}{\mathrm{6}}\:\boldsymbol{{by}}\:\boldsymbol{{using}} \\ $$$$\boldsymbol{{x}}=\mathrm{3}\boldsymbol{{y}}\:\boldsymbol{{and}}\: \\ $$$$\boldsymbol{{sin}}\mathrm{3}\boldsymbol{{y}}=\mathrm{3}\boldsymbol{{siny}}−\mathrm{4}\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{y}} \\ $$

Question Number 71294 Answers: 1 Comments: 1

solve in Z (1/x)+(1/y)=(1/p) with p∈P

$${solve}\:{in}\:\mathbb{Z}\:\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}=\frac{\mathrm{1}}{{p}}\:{with}\:{p}\in\mathbb{P} \\ $$

Question Number 71282 Answers: 4 Comments: 0

solve x(y+z)=27 y(z+x)=32 z(x+y)=35

$${solve} \\ $$$${x}\left({y}+{z}\right)=\mathrm{27} \\ $$$${y}\left({z}+{x}\right)=\mathrm{32} \\ $$$${z}\left({x}+{y}\right)=\mathrm{35} \\ $$

Question Number 71273 Answers: 0 Comments: 1

Question Number 71265 Answers: 0 Comments: 1

Question Number 71255 Answers: 1 Comments: 2

Question Number 71242 Answers: 3 Comments: 0

Given: (a/b) + (c/d) = (b/a) + (d/c) Show that, (a^2 /b^2 ) − (c^2 /d^2 ) = (b^2 /a^2 ) − (d^2 /c^2 )

$$\mathrm{Given}:\:\:\:\frac{\mathrm{a}}{\mathrm{b}}\:+\:\frac{\mathrm{c}}{\mathrm{d}}\:\:=\:\:\frac{\mathrm{b}}{\mathrm{a}}\:+\:\frac{\mathrm{d}}{\mathrm{c}} \\ $$$$\mathrm{Show}\:\mathrm{that},\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{b}^{\mathrm{2}} }\:−\:\frac{\mathrm{c}^{\mathrm{2}} }{\mathrm{d}^{\mathrm{2}} }\:\:=\:\:\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} }\:−\:\frac{\mathrm{d}^{\mathrm{2}} }{\mathrm{c}^{\mathrm{2}} } \\ $$

Question Number 71239 Answers: 2 Comments: 3

∫(1/(2cosx−5sinx−3))dx

$$\int\frac{\mathrm{1}}{\mathrm{2cosx}−\mathrm{5sinx}−\mathrm{3}}\mathrm{dx} \\ $$

Question Number 71235 Answers: 2 Comments: 1

sinh[ln (x + (√(1 + x^2 ))) ] ≡ A. 2x B. (1/x) C. x^2 D. x

$${sinh}\left[{ln}\:\left({x}\:+\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:\right]\:\equiv\: \\ $$$$ \\ $$$${A}.\:\:\mathrm{2}{x} \\ $$$${B}.\:\:\frac{\mathrm{1}}{{x}} \\ $$$${C}.\:\:{x}^{\mathrm{2}} \\ $$$${D}.\:\:{x} \\ $$

Question Number 71233 Answers: 1 Comments: 3

Question Number 71229 Answers: 0 Comments: 1

Question Number 71220 Answers: 0 Comments: 0

Question Number 71216 Answers: 1 Comments: 0

Question Number 71206 Answers: 1 Comments: 0

Let p,q,r are positive real numbers . 0 < r < min{p,q}. Prove that (√(p−r)) + (√(q−r)) ≤ min{(√((pq)/r)) , (√(2(p+q − 2r))) }

$${Let}\:\:{p},{q},{r}\:\:{are}\:\:{positive}\:\:{real}\:\:{numbers}\:. \\ $$$$\mathrm{0}\:<\:{r}\:<\:{min}\left\{{p},{q}\right\}. \\ $$$${Prove}\:\:{that} \\ $$$$\:\:\:\:\:\sqrt{{p}−{r}}\:+\:\sqrt{{q}−{r}}\:\:\leqslant\:\:{min}\left\{\sqrt{\frac{{pq}}{{r}}}\:,\:\sqrt{\mathrm{2}\left({p}+{q}\:−\:\mathrm{2}{r}\right)}\:\right\} \\ $$

Question Number 71196 Answers: 1 Comments: 0

the curve y = f(x), when f(x) is a quadratic expression has a maximum value point at (1,4). The curve touches the line 6x + y = 13. Find the value of x for which y = 8

$${the}\:{curve}\:{y}\:=\:{f}\left({x}\right),\:{when}\:{f}\left({x}\right)\:{is}\:{a}\:{quadratic}\:{expression}\:{has}\: \\ $$$${a}\:{maximum}\:{value}\:{point}\:{at}\:\left(\mathrm{1},\mathrm{4}\right).\:{The}\:{curve}\:{touches}\:{the}\:{line} \\ $$$$\mathrm{6}{x}\:+\:{y}\:=\:\mathrm{13}.\:{Find}\:{the}\:{value}\:{of}\:{x}\:{for}\:{which}\:{y}\:=\:\mathrm{8} \\ $$

Question Number 71184 Answers: 1 Comments: 4

Question Number 71198 Answers: 0 Comments: 0

A particle P is projected from a point O at the edge of a cliff 60m from the sea with a velocity of 30ms^(−1) . When P is at a point B where OB is a horizontal, another particle Qsuch that P and Q hit the sea simultaneously at thesame point A. Gven that they strike the sea 6seconds after P was fired ^ calculate a) the sine of the angle of elevation of projection. b) the distance from A to O. c) the time of flight of Q. d) the Range . (take g = 10ms^(−2) ) please help

$${A}\:{particle}\:{P}\:{is}\:{projected}\:{from}\:\:{a}\:{point}\:{O}\:{at}\:\:{the}\:{edge}\:{of}\:{a}\:{cliff}\:\mathrm{60}{m} \\ $$$${from}\:{the}\:{sea}\:{with}\:{a}\:{velocity}\:{of}\:\mathrm{30}{ms}^{−\mathrm{1}} .\:{When}\:{P}\:{is}\:{at}\:{a}\:{point}\:{B} \\ $$$${where}\:{OB}\:{is}\:{a}\:{horizontal},\:{another}\:{particle}\:{Qsuch}\:{that}\: \\ $$$${P}\:{and}\:{Q}\:{hit}\:{the}\:{sea}\:{simultaneously}\:{at}\:{thesame}\:{point}\:{A}.\:{Gven}\:{that}\:{they} \\ $$$${strike}\:{the}\:{sea}\:\mathrm{6}{seconds}\:{after}\:{P}\:{was}\:{fired}\bar {\:}\:{calculate} \\ $$$$\left.{a}\right)\:{the}\:{sine}\:{of}\:{the}\:{angle}\:{of}\:{elevation}\:{of}\:{projection}. \\ $$$$\left.{b}\right)\:{the}\:{distance}\:{from}\:{A}\:{to}\:{O}. \\ $$$$\left.{c}\right)\:{the}\:{time}\:{of}\:{flight}\:{of}\:{Q}. \\ $$$$\left.{d}\right)\:{the}\:{Range}\:. \\ $$$$\left({take}\:{g}\:=\:\mathrm{10}{ms}^{−\mathrm{2}} \right)\: \\ $$$${please}\:{help}\: \\ $$

Question Number 71197 Answers: 0 Comments: 0