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Question Number 58250 Answers: 1 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3x}^{\mathrm{3}} \:−\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{2x}\:−\:\mathrm{4}}{\sqrt{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3x}\:+\:\mathrm{2}}}\right)\:\mathrm{dx} \\ $$

Question Number 58249 Answers: 0 Comments: 0

I_n ^ =∫_0 ^(π/2) cos^n xcos(nx)dx then show that I_1 ,I_2 ,I_3 ....are in G.P

$$\overset{} {{I}}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{{n}} {xcos}\left({nx}\right){dx} \\ $$$${then}\:{show}\:{that}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,{I}_{\mathrm{3}} ....{are}\:{in}\:{G}.{P} \\ $$

Question Number 58248 Answers: 1 Comments: 0

leg A_1 ,A_2 ,...A_n and H_1 ,H_2 ,...H_n are n A.M′S and H.M′S respectively between a and b prove that A_r H_(n−r+1) =ab n≥r≥1

$${leg}\:{A}_{\mathrm{1}} ,{A}_{\mathrm{2}} ,...{A}_{{n}} \:{and}\:{H}_{\mathrm{1}} ,{H}_{\mathrm{2}} ,...{H}_{{n}} \:{are}\:{n}\:{A}.{M}'{S}\: \\ $$$${and}\:{H}.{M}'{S}\:{respectively}\:{between}\:{a}\:{and}\:{b} \\ $$$${prove}\:{that}\:{A}_{{r}} {H}_{{n}−{r}+\mathrm{1}} ={ab} \\ $$$$\:{n}\geqslant{r}\geqslant\mathrm{1} \\ $$

Question Number 58247 Answers: 0 Comments: 0

lim_(n→∞) ((S_1 S_n +S_2 S_(n−1) +S_3 S_(n−2) +...+S_n S_1 )/(S_1 ^2 +S_2 ^2 +...+S_n ^2 )) when S_n is sum of infinite series whose first term=n and common ratio (1/(n+1)) find the value of limit

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{S}_{\mathrm{1}} {S}_{{n}} +{S}_{\mathrm{2}} {S}_{{n}−\mathrm{1}} +{S}_{\mathrm{3}} {S}_{{n}−\mathrm{2}} +...+{S}_{{n}} {S}_{\mathrm{1}} }{{S}_{\mathrm{1}} ^{\mathrm{2}} +{S}_{\mathrm{2}} ^{\mathrm{2}} +...+{S}_{{n}} ^{\mathrm{2}} } \\ $$$${when}\:{S}_{{n}} \:{is}\:{sum}\:{of}\:{infinite}\:{series}\:{whose} \\ $$$${first}\:{term}={n}\:\:\:{and}\:{common}\:{ratio}\:\frac{\mathrm{1}}{{n}+\mathrm{1}} \\ $$$${find}\:{the}\:{value}\:{of}\:{limit} \\ $$

Question Number 58246 Answers: 1 Comments: 0

show that P=x^(9999) +x^(8888) +x^(7777) +x^(6666) +x^(5555) +x^(4444) +x^(3333) +x^(2222) +x^(1111) +1 Q=x^9 +x^8 +x^7 +x^6 +x^5 +x^4 +x^3 +x^2 +x+1 prove P is divisible by Q

$${show}\:{that} \\ $$$${P}={x}^{\mathrm{9999}} +{x}^{\mathrm{8888}} +{x}^{\mathrm{7777}} +{x}^{\mathrm{6666}} +{x}^{\mathrm{5555}} +{x}^{\mathrm{4444}} +{x}^{\mathrm{3333}} +{x}^{\mathrm{2222}} +{x}^{\mathrm{1111}} +\mathrm{1} \\ $$$${Q}={x}^{\mathrm{9}} +{x}^{\mathrm{8}} +{x}^{\mathrm{7}} +{x}^{\mathrm{6}} +{x}^{\mathrm{5}} +{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1} \\ $$$${prove}\:\:{P}\:\:{is}\:{divisible}\:{by}\:{Q} \\ $$

Question Number 58245 Answers: 1 Comments: 0

if log(a+b+c)=loga+logb+logc prove log(((2a)/(1−a^2 ))+((2b)/(1−b^2 ))+((2c)/(1−c^2 )))=log(((2a)/(1−a^2 )))+log(((2b)/(1−b^2 )))+log(((2c)/(1−c^2 )))

$${if}\:{log}\left({a}+{b}+{c}\right)={loga}+{logb}+{logc} \\ $$$${prove} \\ $$$${log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}−{a}^{\mathrm{2}} }+\frac{\mathrm{2}{b}}{\mathrm{1}−{b}^{\mathrm{2}} }+\frac{\mathrm{2}{c}}{\mathrm{1}−{c}^{\mathrm{2}} }\right)={log}\left(\frac{\mathrm{2}{a}}{\mathrm{1}−{a}^{\mathrm{2}} }\right)+{log}\left(\frac{\mathrm{2}{b}}{\mathrm{1}−{b}^{\mathrm{2}} }\right)+{log}\left(\frac{\mathrm{2}{c}}{\mathrm{1}−{c}^{\mathrm{2}} }\right) \\ $$

Question Number 58240 Answers: 1 Comments: 2

i=∫dx/(ax^2 +bx+c)^(3/2)

$$\mathrm{i}=\int\mathrm{dx}/\left(\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$

Question Number 58239 Answers: 1 Comments: 0

lim_(x→0^+ ) (x^(1/x) )

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

Question Number 58238 Answers: 0 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:+\:\mathrm{2}\boldsymbol{\mathrm{x}}\:−\:\mathrm{4}}{\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:−\:\mathrm{3}\boldsymbol{\mathrm{x}}\:+\:\mathrm{2}}}\right)\:\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 58222 Answers: 2 Comments: 4

∫_0 ^1 x^x dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{x}} {dx} \\ $$

Question Number 58220 Answers: 1 Comments: 0

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

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

Question Number 58216 Answers: 1 Comments: 1

The molar heat capacity of a metal at low temperature varies with the temperature according to the equation C = bθ + (a/H)θ^3 where a, b and H are constant. How much heat per mole is transfered during the process in which the temperature change from 0.01H to 0.02H ?

$$\mathrm{The}\:\mathrm{molar}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{metal}\:\:\mathrm{at} \\ $$$$\mathrm{low}\:\mathrm{temperature}\:\mathrm{varies}\:\mathrm{with}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\mathrm{C}\:=\:\mathrm{b}\theta\:+\:\frac{\mathrm{a}}{\mathrm{H}}\theta^{\mathrm{3}} \\ $$$$\mathrm{where}\:\mathrm{a},\:\mathrm{b}\:\mathrm{and}\:\mathrm{H}\:\mathrm{are}\:\mathrm{constant}. \\ $$$$\mathrm{How}\:\mathrm{much}\:\mathrm{heat}\:\mathrm{per}\:\mathrm{mole}\:\mathrm{is}\:\mathrm{transfered} \\ $$$$\mathrm{during}\:\mathrm{the}\:\mathrm{process}\:\mathrm{in}\:\mathrm{which}\:\mathrm{the}\: \\ $$$$\mathrm{temperature}\:\mathrm{change}\:\mathrm{from}\:\mathrm{0}.\mathrm{01H}\: \\ $$$$\mathrm{to}\:\mathrm{0}.\mathrm{02H}\:? \\ $$

Question Number 58212 Answers: 0 Comments: 0

let f(x) =∫_0 ^∞ e^(−x[t]) sin(xt)dt with x>0 1) find a explicit form for f(x) 2) let U_n =nf(n) find lim_(n→+∞) U_n and study the convergence of ΣU_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} \:{sin}\left({xt}\right){dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{U}_{{n}} ={nf}\left({n}\right)\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{U}_{{n}} \\ $$

Question Number 58211 Answers: 0 Comments: 2

∫_( 0) ^(2π) ∣ cos x−sin x ∣dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\:\mid\:\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\:\mid{dx}\:= \\ $$

Question Number 58210 Answers: 0 Comments: 5

find two possible number such that 1) xy=(x/y)=x−y 2)xy=((2x)/y)=3(x−y) 3) xy=(x/y)=2(x−y).

$$\mathrm{find}\:\mathrm{two}\:\mathrm{possible}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left.\mathrm{1}\right)\:\:\mathrm{xy}=\frac{\mathrm{x}}{\mathrm{y}}=\mathrm{x}−\mathrm{y} \\ $$$$\left.\mathrm{2}\right)\mathrm{xy}=\frac{\mathrm{2x}}{\mathrm{y}}=\mathrm{3}\left(\mathrm{x}−\mathrm{y}\right) \\ $$$$\left.\mathrm{3}\right)\:\:\mathrm{xy}=\frac{\mathrm{x}}{\mathrm{y}}=\mathrm{2}\left(\mathrm{x}−\mathrm{y}\right). \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 58203 Answers: 1 Comments: 0

A battery can supply current of 1.2A and 0.4A through 4Ω and 14Ω respectively. Calculate the internal resistance of the battery

$$\mathrm{A}\:\mathrm{battery}\:\mathrm{can}\:\mathrm{supply}\:\mathrm{current}\:\mathrm{of}\:\mathrm{1}.\mathrm{2A}\: \\ $$$$\mathrm{and}\:\mathrm{0}.\mathrm{4A}\:\mathrm{through}\:\mathrm{4}\Omega\:\mathrm{and}\:\mathrm{14}\Omega\:\:\mathrm{respectively}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{internal}\:\mathrm{resistance}\:\mathrm{of}\:\mathrm{the}\:\mathrm{battery} \\ $$

Question Number 58196 Answers: 1 Comments: 0

The molar heat capacity of constant presure of a gas varies with the temperature according to the equation C_p = a + bθ −(C/θ^2 ) where a,b and C are constants. How much heat is transfered during an isobaric process in which n mole of gas undergo a temperature rise from θ_(i ) to θ_f ?

$$\mathrm{The}\:\mathrm{molar}\:\mathrm{heat}\:\mathrm{capacity}\:\mathrm{of}\:\mathrm{constant} \\ $$$$\mathrm{presure}\:\mathrm{of}\:\mathrm{a}\:\mathrm{gas}\:\mathrm{varies}\:\mathrm{with}\:\mathrm{the}\:\mathrm{temperature} \\ $$$$\mathrm{according}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{C}_{\mathrm{p}} \:=\:\:\mathrm{a}\:+\:\mathrm{b}\theta\:−\frac{\mathrm{C}}{\theta^{\mathrm{2}} } \\ $$$$\mathrm{where}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{constants}. \\ $$$$\:\:\mathrm{How}\:\mathrm{much}\:\mathrm{heat}\:\mathrm{is}\:\mathrm{transfered}\:\mathrm{during} \\ $$$$\:\:\:\mathrm{an}\:\mathrm{isobaric}\:\mathrm{process}\:\mathrm{in}\:\mathrm{which}\:\mathrm{n}\:\mathrm{mole} \\ $$$$\:\:\:\mathrm{of}\:\mathrm{gas}\:\mathrm{undergo}\:\mathrm{a}\:\mathrm{temperature}\:\mathrm{rise} \\ $$$$\:\:\:\:\mathrm{from}\:\theta_{{i}\:} \mathrm{to}\:\theta_{{f}} \:? \\ $$

Question Number 58195 Answers: 2 Comments: 0

(1/x)+(1/y)=(3/4) (x^2 /y)+(y^2 /x)=9 find the value of x and y

$$\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{y}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}}+\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{x}}=\mathrm{9} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$

Question Number 58193 Answers: 0 Comments: 0

If z ∈ C such that R(z^n )>0 for n∈N^+ . Show that z ∈R^+ .

$$\mathrm{If}\:{z}\:\in\:\mathbb{C}\:\:\mathrm{such}\:\mathrm{that}\:\mathfrak{R}\left({z}^{{n}} \right)>\mathrm{0}\:\mathrm{for}\:{n}\in\mathbb{N}^{+} . \\ $$$$\mathrm{Show}\:\mathrm{that}\:{z}\:\in\mathbb{R}^{+} . \\ $$

Question Number 58187 Answers: 0 Comments: 0

let f(x) =∫_1 ^3 arctan(x+(x/t))dt withx>0 1) determine a explicit form of f(x) 2) give f^′ (x) at form of integral and find its value 3) calculate ∫_1 ^3 arctan(1+(1/t))dt and ∫_1 ^3 arctan(2+(2/t))dt . 4) calculate ∫_1 ^3 (2t−1)arctan(1+(1/t))dt .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left({x}+\frac{{x}}{{t}}\right){dt}\:\:\:{withx}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{give}\:{f}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{find}\:{its}\:{value} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:\:\:{and}\:\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{2}+\frac{\mathrm{2}}{{t}}\right){dt}\:. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\left(\mathrm{2}{t}−\mathrm{1}\right){arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:. \\ $$

Question Number 58185 Answers: 0 Comments: 0

find ∫ ((xdx)/(cosx +sin(2x)))

$${find}\:\int\:\:\:\frac{{xdx}}{{cosx}\:+{sin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 58184 Answers: 0 Comments: 0

find ∫ ((xdx)/(sinx +cos(2x)))

$${find}\:\:\int\:\:\:\:\:\:\frac{{xdx}}{{sinx}\:+{cos}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 58178 Answers: 0 Comments: 0

Question Number 58177 Answers: 2 Comments: 0

cos^(−1) (1/2) + 2 sin^(−1) (1/2) =

$$\mathrm{cos}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{2}\:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\:\:= \\ $$

Question Number 58176 Answers: 1 Comments: 0

If x_1 , x_2 , x_3 , x_4 are roots of the equation x^4 −x^3 sin 2β+x^2 cos 2β−x cos β−sin β=0, then tan^(−1) x_1 +tan^(−1) x_2 +tan^(−1) x_3 +tan^(−1) x_4 =

$$\mathrm{If}\:\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:{x}_{\mathrm{3}} ,\:{x}_{\mathrm{4}} \:\:\mathrm{are}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${x}^{\mathrm{4}} −{x}^{\mathrm{3}} \mathrm{sin}\:\mathrm{2}\beta+{x}^{\mathrm{2}} \mathrm{cos}\:\mathrm{2}\beta−{x}\:\mathrm{cos}\:\beta−\mathrm{sin}\:\beta=\mathrm{0}, \\ $$$$\mathrm{then} \\ $$$$\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{1}} +\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{2}} +\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{3}} +\mathrm{tan}^{−\mathrm{1}} {x}_{\mathrm{4}} = \\ $$

Question Number 58175 Answers: 1 Comments: 0

The value of sin(π+θ) sin(π−θ) cosec^2 θ is equal to

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{sin}\left(\pi+\theta\right)\:\mathrm{sin}\left(\pi−\theta\right)\:\mathrm{cosec}^{\mathrm{2}} \theta \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

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