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Question Number 57487 Answers: 0 Comments: 1

calculate lim_(x→1) ∫_x ^x^2 ((arctan(t))/(sint))dt .

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\:\frac{{arctan}\left({t}\right)}{{sint}}{dt}\:. \\ $$

Question Number 57486 Answers: 0 Comments: 4

let f(x) =((ln(1+x))/(2−x^2 )) 1)calculate f^((n)) (x) 2) calculate f^((n)) (0) 3)developp f(x) at integr serie.

$${let}\:{f}\left({x}\right)\:=\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{2}−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:\: \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right){developp}\:{f}\left({x}\right)\:{at}\:{integr}\:{serie}. \\ $$

Question Number 57480 Answers: 0 Comments: 2

if F(x,y)=F(y,x) and x+y=c (constant) prove that F_(max or min) =F((c/2),(c/2)).

$${if}\:{F}\left({x},{y}\right)={F}\left({y},{x}\right)\:{and}\:{x}+{y}={c}\:\left({constant}\right) \\ $$$${prove}\:{that}\:{F}_{{max}\:{or}\:{min}} ={F}\left(\frac{{c}}{\mathrm{2}},\frac{{c}}{\mathrm{2}}\right). \\ $$

Question Number 57474 Answers: 1 Comments: 2

prove sin18×cos36=(1/4)

$$\mathrm{prove}\: \\ $$$$\boldsymbol{{sin}}\mathrm{18}×\boldsymbol{{cos}}\mathrm{36}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Question Number 57469 Answers: 0 Comments: 3

((4tan75)/(1−tan^2 75))=(1/(cos150)) find tan75 in surd form

$$\frac{\mathrm{4}{tan}\mathrm{75}}{\mathrm{1}−{tan}^{\mathrm{2}} \mathrm{75}}=\frac{\mathrm{1}}{{cos}\mathrm{150}}\: \\ $$$${find}\:{tan}\mathrm{75}\:{in}\:{surd}\:{form} \\ $$

Question Number 57462 Answers: 0 Comments: 0

Two conductors has total charge of +10.0μC and −10μC with 10volt between them. (a) Determine the capacitance between them (b) what is the p.d between the two condoctors if the charge on each are increased to +100μC and −100μC respectively ?

$$\mathrm{Two}\:\mathrm{conductors}\:\mathrm{has}\:\mathrm{total}\:\mathrm{charge}\:\mathrm{of} \\ $$$$+\mathrm{10}.\mathrm{0}\mu\mathrm{C}\:\mathrm{and}\:−\mathrm{10}\mu\mathrm{C}\:\mathrm{with}\:\mathrm{10volt}\: \\ $$$$\mathrm{between}\:\mathrm{them}. \\ $$$$\:\left(\mathrm{a}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{capacitance}\:\mathrm{between}\:\mathrm{them} \\ $$$$\:\left(\mathrm{b}\right)\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{p}.\mathrm{d}\:\mathrm{between}\:\mathrm{the}\:\mathrm{two} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{condoctors}\:\mathrm{if}\:\mathrm{the}\:\mathrm{charge}\:\mathrm{on}\:\mathrm{each} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{are}\:\mathrm{increased}\:\mathrm{to}\:+\mathrm{100}\mu\mathrm{C}\:\mathrm{and}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:−\mathrm{100}\mu\mathrm{C}\:\mathrm{respectively}\:? \\ $$

Question Number 57456 Answers: 0 Comments: 0

Question Number 57448 Answers: 1 Comments: 0

If sin x+cosec x=2, then sin^n x+cosec^n x is equal to

$$\mathrm{If}\:\:\:\mathrm{sin}\:{x}+\mathrm{cosec}\:{x}=\mathrm{2},\:\mathrm{then}\:\mathrm{sin}^{{n}} {x}+\mathrm{cosec}^{{n}} {x} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 57442 Answers: 2 Comments: 4

1) lim_(x→0) (x/(e^(1/x) +1)) = ? 2) For xεR, f(x)=∣ln2−sin x∣ and g(x)=f(f(x)), then prove that g′(0)=cos (ln2).

$$\left.\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}}{{e}^{\frac{\mathrm{1}}{{x}}} +\mathrm{1}}\:=\:? \\ $$$$\left.\mathrm{2}\right)\:{For}\:{x}\epsilon{R},\:{f}\left({x}\right)=\mid{ln}\mathrm{2}−\mathrm{sin}\:{x}\mid\:{and}\: \\ $$$${g}\left({x}\right)={f}\left({f}\left({x}\right)\right),\:{then}\:{prove}\:{that}\: \\ $$$${g}'\left(\mathrm{0}\right)=\mathrm{cos}\:\left({ln}\mathrm{2}\right). \\ $$

Question Number 57439 Answers: 0 Comments: 1

Find all solutions of x, y, z integers that satisfy x^3 + y^3 + z^3 = 33

$${Find}\:\:{all}\:\:{solutions}\:\:{of}\:\:{x},\:{y},\:{z}\:\:\:{integers}\:\:{that}\:\:{satisfy} \\ $$$$\:\:\:\:\:\:\:{x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \:\:=\:\:\mathrm{33} \\ $$

Question Number 57435 Answers: 0 Comments: 0

Question Number 57434 Answers: 0 Comments: 0

is there a way to find the sum to infinity of a product operator e.g product of 1.2.3.4.5 ... [1, infinity]

$$\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{way}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{to}\:\mathrm{infinity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{product}\:\mathrm{operator} \\ $$$$\:\:\mathrm{e}.\mathrm{g}\:\:\:\:\:\mathrm{product}\:\mathrm{of}\:\:\:\:\:\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}.\mathrm{5}\:...\:\:\left[\mathrm{1},\:\mathrm{infinity}\right] \\ $$

Question Number 57433 Answers: 2 Comments: 0

Question Number 57423 Answers: 0 Comments: 0

let A_n =∫_0 ^∞ (dt/((e^t +e^(−t) )^n )) calculate A_n interms of n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({e}^{{t}} \:+\overset{−{t}} {{e}}\right)^{{n}} } \\ $$$${calculate}\:{A}_{{n}} \:{interms}\:{of}\:{n} \\ $$

Question Number 57422 Answers: 0 Comments: 0

let U_n =n ∫_1 ^π ((sinx)/x^n )dx calculate lim_(n→+∞) U_n

$${let}\:{U}_{{n}} ={n}\:\int_{\mathrm{1}} ^{\pi} \:\frac{{sinx}}{{x}^{{n}} }{dx} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 57421 Answers: 1 Comments: 0

calculate ∫_(−1) ^1 (((x^4 +x^2 +1)^2 +e^x )/(e^x +1))dx

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

Question Number 57420 Answers: 0 Comments: 1

let J(x)=∫_0 ^x (t^2 /((√(t+1)) +(√(t+4))))dt find a explicit form of J(x)

$${let}\:{J}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:\:\:\:\frac{{t}^{\mathrm{2}} }{\sqrt{{t}+\mathrm{1}}\:+\sqrt{{t}+\mathrm{4}}}{dt} \\ $$$${find}\:{a}\:{explicit}\:{form}\:{of}\:{J}\left({x}\right) \\ $$

Question Number 57419 Answers: 0 Comments: 1

find ∫_0 ^1 (x+1) ln(x+(√(1+x^2 )))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}+\mathrm{1}\right)\:{ln}\left({x}+\sqrt{\left.\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\right. \\ $$

Question Number 57418 Answers: 0 Comments: 1

calculate ∫_(−1) ^4 ((∣x−1∣+∣x−2∣)/(∣x^2 −9∣ +x^2 +16))dx

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{4}} \:\frac{\mid{x}−\mathrm{1}\mid+\mid{x}−\mathrm{2}\mid}{\mid{x}^{\mathrm{2}} −\mathrm{9}\mid\:+{x}^{\mathrm{2}} \:+\mathrm{16}}{dx} \\ $$

Question Number 57417 Answers: 0 Comments: 2

let F(x) =∫_0 ^x ((1+sint)/(2+cost))dt 1) find a explicite form of f(x) 2) calculate ∫_0 ^π ((1+sint)/(2+cost))dt

$${let}\:{F}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$

Question Number 57416 Answers: 0 Comments: 1

let f(x)=∫_(2x) ^(4x) (dt/(t^2 −2t +3)) 1)find f(x) 2) calculate lim_(x→0) f(x) and lim_(x→+∞) f(x)

$${let}\:{f}\left({x}\right)=\int_{\mathrm{2}{x}} ^{\mathrm{4}{x}} \:\:\:\:\frac{{dt}}{{t}^{\mathrm{2}} −\mathrm{2}{t}\:+\mathrm{3}} \\ $$$$\left.\mathrm{1}\right){find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right) \\ $$

Question Number 57415 Answers: 0 Comments: 1

solve (x−1)y^′ +(1+(√x))y =x e^(−2x)

$${solve}\:\left({x}−\mathrm{1}\right){y}^{'} \:+\left(\mathrm{1}+\sqrt{{x}}\right){y}\:={x}\:{e}^{−\mathrm{2}{x}} \\ $$

Question Number 57414 Answers: 0 Comments: 2

solve y′ =2y^2 +y and y(o)=1

$${solve}\:\:{y}'\:=\mathrm{2}{y}^{\mathrm{2}} \:+{y}\:\:\:{and}\:{y}\left({o}\right)=\mathrm{1} \\ $$

Question Number 57413 Answers: 0 Comments: 0

prove that ln(1+x)>((arctanx)/(1+x)) ∀x>0

$${prove}\:{that}\:{ln}\left(\mathrm{1}+{x}\right)>\frac{{arctanx}}{\mathrm{1}+{x}}\:\:\forall{x}>\mathrm{0} \\ $$

Question Number 57412 Answers: 0 Comments: 1

let u_n =1 +(1/(√2)) +(1/(√3)) +...+(1/(√n)) prove that (u_n ) is divdrgente.

$${let}\:{u}_{{n}} =\mathrm{1}\:+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\:+...+\frac{\mathrm{1}}{\sqrt{{n}}} \\ $$$${prove}\:{that}\:\left({u}_{{n}} \right)\:{is}\:{divdrgente}. \\ $$

Question Number 57411 Answers: 1 Comments: 1

let f(x)=arctan((√x)+(√(x+1))) find f^(−1) (x) .

$${let}\:{f}\left({x}\right)={arctan}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right) \\ $$$${find}\:{f}^{−\mathrm{1}} \left({x}\right)\:. \\ $$

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