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Question Number 105415 Answers: 0 Comments: 0

Question Number 105414 Answers: 0 Comments: 2

Question Number 105413 Answers: 0 Comments: 0

Question Number 105412 Answers: 2 Comments: 0

∫_0 ^(π/2) ln (cos x) dx

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

Question Number 105411 Answers: 1 Comments: 0

∫((x^2 +3)/(x^6 (x^2 +1)))dx Is there any special method of decomposition other than the use of partial fractions ?

$$\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{6}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\mathrm{d}{x} \\ $$$${I}\mathrm{s}\:{there}\:{any}\:{special}\:{method}\:{of}\:{decomposition} \\ $$$${other}\:{than}\:{the}\:{use}\:{of}\:{partial}\:{fractions}\:? \\ $$

Question Number 105410 Answers: 2 Comments: 0

lim_(x→0) ((x(1+a cos x)−bsin x)/x^5 ) = 1 find a & b

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\left(\mathrm{1}+{a}\:\mathrm{cos}\:{x}\right)−{b}\mathrm{sin}\:{x}}{{x}^{\mathrm{5}} }\:=\:\mathrm{1} \\ $$$${find}\:{a}\:\&\:{b}\: \\ $$

Question Number 105406 Answers: 0 Comments: 0

Question Number 105404 Answers: 1 Comments: 0

Cube ABCD.EFGH with length side 2 cm. Point P is the center of ABFE plane. The distance of HP line and the BG line is __

$${Cube}\:{ABCD}.{EFGH}\:{with} \\ $$$${length}\:{side}\:\mathrm{2}\:{cm}.\:{Point}\:{P}\:{is}\:{the} \\ $$$${center}\:{of}\:{ABFE}\:{plane}.\:{The} \\ $$$${distance}\:{of}\:{HP}\:{line}\:{and}\:{the}\:{BG} \\ $$$${line}\:{is}\:\_\_\: \\ $$

Question Number 105399 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) ((x^2 dx)/((x^2 +x+1)^2 (2x^2 +3)))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2x}^{\mathrm{2}} \:+\mathrm{3}\right)} \\ $$

Question Number 105393 Answers: 1 Comments: 0

∫_(−π) ^π ((x sin x dx)/((1+x+(√(1+x^2 )))(√(3+sin^2 x))))

$$\underset{−\pi} {\overset{\pi} {\int}}\:\frac{{x}\:\mathrm{sin}\:{x}\:{dx}}{\left(\mathrm{1}+{x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)\sqrt{\mathrm{3}+\mathrm{sin}\:^{\mathrm{2}} {x}}} \\ $$

Question Number 105388 Answers: 3 Comments: 0

Question Number 105386 Answers: 2 Comments: 0

∫ (x^3 /(√((a^2 +x^2 )^3 ))) dx

$$\int\:\frac{{x}^{\mathrm{3}} }{\sqrt{\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)^{\mathrm{3}} }}\:{dx}\: \\ $$

Question Number 105383 Answers: 1 Comments: 0

Given { ((lim_(x→5) ((f(x)−a)/(x−5)) = 8)),((lim_(x→5) ((x^2 −ax+b)/(f(x)−a)) = 1)) :} find the value of b+23

$$\mathcal{G}{iven}\:\begin{cases}{\underset{{x}\rightarrow\mathrm{5}} {\mathrm{lim}}\frac{{f}\left({x}\right)−{a}}{{x}−\mathrm{5}}\:=\:\mathrm{8}}\\{\underset{{x}\rightarrow\mathrm{5}} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} −{ax}+{b}}{{f}\left({x}\right)−{a}}\:=\:\mathrm{1}}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:{b}+\mathrm{23}\: \\ $$

Question Number 105380 Answers: 1 Comments: 1

Question Number 105366 Answers: 1 Comments: 0

Question Number 105370 Answers: 4 Comments: 0

Question Number 105361 Answers: 0 Comments: 0

solve by Frobenius method x^2 y′′−x(2x−1)y′+(x+1)y=0

$${solve}\:{by}\:{Frobenius}\:{method} \\ $$$${x}^{\mathrm{2}} {y}''−{x}\left(\mathrm{2}{x}−\mathrm{1}\right){y}'+\left({x}+\mathrm{1}\right){y}=\mathrm{0} \\ $$

Question Number 105356 Answers: 1 Comments: 0

Given a cube ABCD.EFGH. if α is the angle BEH & BEG , find the value of cos α ?

$$\mathcal{G}{iven}\:{a}\:{cube}\:{ABCD}.{EFGH}. \\ $$$${if}\:\alpha\:{is}\:{the}\:{angle}\:{BEH}\:\&\:{BEG}\:, \\ $$$${find}\:{the}\:{value}\:{of}\:\mathrm{cos}\:\alpha\:? \\ $$

Question Number 105353 Answers: 0 Comments: 0

solve y^(′′ ) +2y^′ −y =x^n e^(−x) n integr natural

$${solve}\:{y}^{''\:} +\mathrm{2}{y}^{'} −{y}\:={x}^{{n}} \:{e}^{−{x}} \\ $$$${n}\:{integr}\:{natural} \\ $$

Question Number 105340 Answers: 1 Comments: 0

1\Show that the polynomial, P(x)=x^n +ax+b, n≥2 (a,b)∈R^2 has at most 3 real distinct roots. 2\Show that the equation x^4 +4ax+b=0 has no more than 2 real distinct roots.

$$\mathrm{1}\backslash\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{polynomial},\:\mathrm{P}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}} +\mathrm{ax}+\mathrm{b},\:\mathrm{n}\geqslant\mathrm{2} \\ $$$$\left(\mathrm{a},\mathrm{b}\right)\in\mathbb{R}^{\mathrm{2}} \:\mathrm{has}\:\mathrm{at}\:\mathrm{most}\:\mathrm{3}\:\mathrm{real}\:\mathrm{distinct}\:\mathrm{roots}. \\ $$$$\mathrm{2}\backslash\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{4}} +\mathrm{4ax}+\mathrm{b}=\mathrm{0}\:\mathrm{has}\:\mathrm{no} \\ $$$$\mathrm{more}\:\mathrm{than}\:\mathrm{2}\:\mathrm{real}\:\mathrm{distinct}\:\mathrm{roots}. \\ $$

Question Number 105331 Answers: 0 Comments: 3

lim_(t→0) (1/t)ln[1−((ln(1+t))/(ln(t)))]

$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{t}}\mathrm{ln}\left[\mathrm{1}−\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{t}\right)}{\mathrm{ln}\left(\mathrm{t}\right)}\right] \\ $$

Question Number 105327 Answers: 2 Comments: 3

If x = ((11)/8) and y = ((66)/9) then what is the answer of ((x+y)/(x−y)) ?

$$\mathrm{If}\:\mathrm{x}\:=\:\frac{\mathrm{11}}{\mathrm{8}}\:\mathrm{and}\:\mathrm{y}\:=\:\frac{\mathrm{66}}{\mathrm{9}}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{answer}\:\mathrm{of}\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{x}−\mathrm{y}}\:?\: \\ $$

Question Number 105322 Answers: 0 Comments: 1

Question Number 105320 Answers: 0 Comments: 0

find C kx^2 +ky^2 +z^2 ≥C x,y,z,k∈R such that xy+yz+zx=1

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{C}} \\ $$$${kx}^{\mathrm{2}} +{ky}^{\mathrm{2}} +{z}^{\mathrm{2}} \geqslant{C} \\ $$$${x},{y},{z},{k}\in\mathbb{R}\:\:{such}\:{that} \\ $$$${xy}+{yz}+{zx}=\mathrm{1} \\ $$

Question Number 105234 Answers: 0 Comments: 0

f:R→R x,y∈R f(f(x)f(y))+f(x+y)=f(xy) f(x)=?

$${f}:\mathbb{R}\rightarrow\mathbb{R}\:\:{x},{y}\in\mathbb{R} \\ $$$${f}\left({f}\left({x}\right){f}\left({y}\right)\right)+{f}\left({x}+{y}\right)={f}\left({xy}\right) \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 105233 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((ch(cosx)−cos(chx))/(x^2 +3))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{ch}\left(\mathrm{cosx}\right)−\mathrm{cos}\left(\mathrm{chx}\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{3}}\mathrm{dx} \\ $$

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