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Question Number 87817 Answers: 1 Comments: 3

f(((x−3)/(x+1)))+f(((x+3)/(x−1)))=x find f(x)

$${f}\left(\frac{{x}−\mathrm{3}}{{x}+\mathrm{1}}\right)+{f}\left(\frac{{x}+\mathrm{3}}{{x}−\mathrm{1}}\right)={x} \\ $$$${find}\:{f}\left({x}\right) \\ $$

Question Number 87815 Answers: 1 Comments: 3

I = ∫_0 ^(π/2) cos 2x(cos^4 x+sin^4 x) dx

$$\mathrm{I}\:=\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{cos}\:\mathrm{2x}\left(\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}+\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 87803 Answers: 2 Comments: 4

Question Number 87799 Answers: 0 Comments: 2

f(x)= { ((ax^2 +bx −1≤x≤0)),((cx^2 +d 0<x≤(1/2))),((bx+d (1/2)<x≤1)) :} f(x) is continuous on[−1,1] prove d=0 c=2b

$${f}\left({x}\right)=\begin{cases}{{ax}^{\mathrm{2}} +{bx}\:\:\:\:\:\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{0}}\\{{cx}^{\mathrm{2}} +{d}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}<{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}}}\\{{bx}+{d}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}<{x}\leqslant\mathrm{1}}\end{cases} \\ $$$${f}\left({x}\right)\:{is}\:{continuous}\:{on}\left[−\mathrm{1},\mathrm{1}\right] \\ $$$${prove}\:{d}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{c}=\mathrm{2}{b} \\ $$

Question Number 87793 Answers: 2 Comments: 0

show that ∫e^(sin(x)) dx= −Σ_(n=0) ^∞ (1/(n!))[ cos(x)∗(sin(x))^(n+1) ∗[(sin(x))^2 ]^((((−n)/2)−(1/2))) ∗ 2F_1 [(1/2),((1−n)/2);(3/2);(cos(x))^2 ] ]+c notice\2F_1 is special function called hypergeometric function

$${show}\:{that} \\ $$$$\int{e}^{{sin}\left({x}\right)} \:{dx}= \\ $$$$−\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\left[\:{cos}\left({x}\right)\ast\left({sin}\left({x}\right)\right)^{{n}+\mathrm{1}} \ast\left[\left({sin}\left({x}\right)\right)^{\mathrm{2}} \right]^{\left(\frac{−{n}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\right)} \ast\:\mathrm{2}{F}_{\mathrm{1}} \left[\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}−{n}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};\left({cos}\left({x}\right)\right)^{\mathrm{2}} \right]\:\right]+{c} \\ $$$$ \\ $$$${notice}\backslash\mathrm{2}{F}_{\mathrm{1}} \:{is}\:{special}\:{function}\:{called}\:{hypergeometric}\:{function} \\ $$

Question Number 87790 Answers: 2 Comments: 0

How many handshakes are exchanged betwen 27 boys?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{handshakes}\:\mathrm{are}\:\mathrm{exchanged} \\ $$$$\mathrm{betwen}\:\mathrm{27}\:\mathrm{boys}? \\ $$

Question Number 87789 Answers: 1 Comments: 1

(D^3 −D^2 )y = x^2 +1 ,y(0)=1 y ′(0)=−1 ,y ′′(0) = 0

$$\left(\mathrm{D}^{\mathrm{3}} −\mathrm{D}^{\mathrm{2}} \right)\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{1}\:,\mathrm{y}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$$\mathrm{y}\:'\left(\mathrm{0}\right)=−\mathrm{1}\:,\mathrm{y}\:''\left(\mathrm{0}\right)\:=\:\mathrm{0} \\ $$

Question Number 87785 Answers: 0 Comments: 0

Question Number 87784 Answers: 1 Comments: 1

Given f(x,y) = y f(y,x) +x find the value of f(1,2) .

$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\mathrm{y}\:\mathrm{f}\left(\mathrm{y},\mathrm{x}\right)\:+\mathrm{x} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{1},\mathrm{2}\right)\:. \\ $$

Question Number 87773 Answers: 0 Comments: 4

^3 log (^x^2 log (^x^2 log x^4 ))> 0

$$\:^{\mathrm{3}} \mathrm{log}\:\left(\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\left(\:^{\mathrm{x}^{\mathrm{2}} } \mathrm{log}\:\mathrm{x}^{\mathrm{4}} \right)\right)>\:\mathrm{0} \\ $$

Question Number 87769 Answers: 2 Comments: 0

∫ ((ln(e^x +1))/(e^(−x) +1)) dx

$$\int\:\frac{\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} +\mathrm{1}\right)}{\mathrm{e}^{−\mathrm{x}} +\mathrm{1}}\:\mathrm{dx}\: \\ $$

Question Number 87767 Answers: 0 Comments: 1

Question Number 87759 Answers: 0 Comments: 2

Question Number 87757 Answers: 0 Comments: 2

∫_a ^b ((√(x−a))/(√(b−x))) dx =?

$$\int_{\mathrm{a}} ^{\mathrm{b}} \:\frac{\sqrt{\mathrm{x}−\mathrm{a}}}{\sqrt{\mathrm{b}−\mathrm{x}}}\:\mathrm{dx}\:=?\: \\ $$

Question Number 87755 Answers: 0 Comments: 2

f(((x−3)/(x+1))) + f(((x+3)/(1−x))) = x find f(x)

$$\mathrm{f}\left(\frac{\mathrm{x}−\mathrm{3}}{\mathrm{x}+\mathrm{1}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{1}−\mathrm{x}}\right)\:=\:\mathrm{x} \\ $$$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$

Question Number 87754 Answers: 0 Comments: 0

Given that forces F_(1 ) and F_2 position vectors r_(1 ) and r_2 F_1 = (2i + 3j)N r_1 = i + 2j F_2 = (αi−7j) N r_2 = 3i + 4j Given that these system of forces form a couple find the value of α.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{forces}\:\mathrm{F}_{\mathrm{1}\:} \:\mathrm{and}\:\mathrm{F}_{\mathrm{2}} \:\mathrm{position}\:\mathrm{vectors}\:\mathrm{r}_{\mathrm{1}\:} \mathrm{and}\:\mathrm{r}_{\mathrm{2}} \\ $$$$\:\:\boldsymbol{\mathrm{F}}_{\mathrm{1}} \:=\:\left(\mathrm{2}\boldsymbol{{i}}\:+\:\mathrm{3}\boldsymbol{{j}}\right)\mathrm{N}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} =\:\boldsymbol{\mathrm{i}}\:+\:\mathrm{2}\boldsymbol{\mathrm{j}} \\ $$$$\:\:\:\boldsymbol{\mathrm{F}}_{\mathrm{2}} \:=\:\left(\alpha\boldsymbol{{i}}−\mathrm{7}\boldsymbol{{j}}\right)\:\mathrm{N}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:=\:\mathrm{3}\boldsymbol{\mathrm{i}}\:+\:\mathrm{4}\boldsymbol{\mathrm{j}} \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{these}\:\mathrm{system}\:\mathrm{of}\:\mathrm{forces}\:\mathrm{form}\:\mathrm{a}\:\mathrm{couple} \\ $$$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\alpha. \\ $$

Question Number 87752 Answers: 1 Comments: 0

A particle exhibits simple hamornic motion such that (d^2 x/dt^2 ) + 4x = 0 Calculate the period of the ocsillation

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{exhibits}\:\mathrm{simple}\:\mathrm{hamornic}\:\mathrm{motion}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }\:+\:\mathrm{4}{x}\:=\:\mathrm{0} \\ $$$$\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{period}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ocsillation}\: \\ $$

Question Number 87751 Answers: 0 Comments: 2

find in the form y= f(x) the general solution of the differentail equation (d^2 y/dx^2 ) −(dy/dx)−6y = e^(3x)

$$\mathrm{find}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:{y}=\:{f}\left({x}\right)\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{differentail}\:\mathrm{equation} \\ $$$$\:\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:−\frac{{dy}}{{dx}}−\mathrm{6}{y}\:=\:{e}^{\mathrm{3}{x}} \\ $$$$ \\ $$

Question Number 87737 Answers: 1 Comments: 0

solve sin((π/([(([x])/4)])))=(1/2)

$${solve} \\ $$$${sin}\left(\frac{\pi}{\left[\frac{\left[{x}\right]}{\mathrm{4}}\right]}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 87733 Answers: 1 Comments: 2