Question and Answers Forum

let j =e^((i2π)/3) and p(x) =(1+xj)^n −(1−xj)^n 1) find the roots of p(x) and factorize inside C[x] p(x) 2) decompose inside C(x) the fration F(x) =(1/(p(x))) .

$${let}\:{j}\:={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:\:\:{p}\left({x}\right)\:=\left(\mathrm{1}+{xj}\right)^{{n}} \:−\left(\mathrm{1}−{xj}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:{and}\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fration}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{p}\left({x}\right)}\:. \\ $$

Question Number 42506 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((ln(t))/((^3 (√t^2 ))(1+t)))dt .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({t}\right)}{\left(^{\mathrm{3}} \sqrt{{t}^{\mathrm{2}} }\right)\left(\mathrm{1}+{t}\right)}{dt}\:. \\ $$

Question Number 42505 Answers: 0 Comments: 2

calculate ∫_0 ^∞ ((ln(t))/((1+t)(√t))) dt .

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{{t}}}\:{dt}\:. \\ $$

Question Number 42504 Answers: 0 Comments: 1

let x>0 prove that ∫_0 ^∞ ((e^(−t^2 ) ln(1+xt^2 ))/t^2 ) dt =π ∫_0 ^(√x) e^(1/u^2 ) du .

$${let}\:{x}>\mathrm{0}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{t}^{\mathrm{2}} } {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }\:{dt}\:=\pi\:\int_{\mathrm{0}} ^{\sqrt{{x}}} \:\:{e}^{\frac{\mathrm{1}}{{u}^{\mathrm{2}} }} \:\:{du}\:. \\ $$

Question Number 42503 Answers: 1 Comments: 1

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

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

Question Number 42502 Answers: 0 Comments: 0

calculate A_n = ∫_0 ^(2π) ((cos(nx))/(cosx +sinx))dx with n from N

$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cos}\left({nx}\right)}{{cosx}\:+{sinx}}{dx}\:\:{with}\:{n}\:{from}\:{N} \\ $$

Question Number 42501 Answers: 0 Comments: 1

calculate I = ∫_0 ^(2π) ((cos(4x))/(cosx +sinx)) and J = ∫_0 ^(2π) ((sin(4x))/(cosx +sinx))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{4}{x}\right)}{{cosx}\:+{sinx}}\:\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{4}{x}\right)}{{cosx}\:+{sinx}}{dx} \\ $$

Question Number 42500 Answers: 0 Comments: 2

calculate I = ∫_0 ^(2π) ((cos(2x))/(cosx +sinx))dx and J =∫_0 ^(2π) ((sin(2x))/(cosx +sinx))dx

$${calculate}\:\:\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{{cosx}\:+{sinx}}{dx}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{cosx}\:+{sinx}}{dx} \\ $$

Question Number 42497 Answers: 0 Comments: 1

A body cools from 90°C to 40°C in 2 minutes at a temperature,20°C of the surrounding.Calculate the temperature of the body after another 5 minutes.

$${A}\:{body}\:{cools}\:{from}\:\mathrm{90}°{C}\:{to}\:\mathrm{40}°{C}\:{in} \\ $$$$\mathrm{2}\:{minutes}\:{at}\:{a}\:{temperature},\mathrm{20}°{C} \\ $$$${of}\:{the}\:{surrounding}.{Calculate}\:{the} \\ $$$${temperature}\:{of}\:{the}\:{body}\:{after} \\ $$$${another}\:\mathrm{5}\:{minutes}. \\ $$

Question Number 42495 Answers: 0 Comments: 0

In the sequence 1, 22, 333, ... 10101010101010101010, 1111111111111111111111, ... The sum of the digits in the 200th term is ??

$$\mathrm{In}\:\mathrm{the}\:\mathrm{sequence}\:\:\mathrm{1},\:\mathrm{22},\:\mathrm{333},\:...\:\mathrm{10101010101010101010},\:\mathrm{1111111111111111111111},\:... \\ $$$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{the}\:\mathrm{200th}\:\mathrm{term}\:\mathrm{is}\:?? \\ $$

Question Number 42494 Answers: 1 Comments: 1

In the sequence of numbers 1, 2, 11, 22, 111, 222, ... the sum of the digits in 999th terms is ??

$$\mathrm{In}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{numbers}\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{11},\:\mathrm{22},\:\mathrm{111},\:\mathrm{222},\:...\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{in}\:\mathrm{999th}\:\mathrm{terms}\:\mathrm{is}\:?? \\ $$

Question Number 42493 Answers: 0 Comments: 1

calculate lim_(n→+∞) Σ_(1≤i<j≤n) (1/(i^x j^x )) with x>1 for that consider ξ(x) =Σ_(n=1) ^∞ (1/n^x ) 2) calculate lim_(n→+∞) Σ_(1≤i<j≤n) (1/((ij)^2 )) .

$$\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\:\:\frac{\mathrm{1}}{{i}^{{x}} {j}^{{x}} }\:\:\:{with}\:\:{x}>\mathrm{1}\:\:{for}\:{that}\:{consider} \\ $$$$\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{{x}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\:\:\:\frac{\mathrm{1}}{\left({ij}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 42492 Answers: 2 Comments: 0

let x>0 ,y>0,z>0 prove that (x^2 /(yz)) +(y^2 /(xz)) +(z^2 /(xy)) ≥3 .

$${let}\:{x}>\mathrm{0}\:,{y}>\mathrm{0},{z}>\mathrm{0}\:\:\:{prove}\:{that}\:\:\frac{{x}^{\mathrm{2}} }{{yz}}\:+\frac{{y}^{\mathrm{2}} }{{xz}}\:+\frac{{z}^{\mathrm{2}} }{{xy}}\:\geqslant\mathrm{3}\:. \\ $$

Question Number 42490 Answers: 0 Comments: 0

find L(arctanx) .L means laplace transform .

$${find}\:{L}\left({arctanx}\right)\:\:.{L}\:{means}\:{laplace}\:{transform}\:. \\ $$

Question Number 42489 Answers: 0 Comments: 1

calculate L (sinxe^(−ax) ) with a>0 L means laplace transform .

$${calculate}\:{L}\:\left({sinxe}^{−{ax}} \right)\:\:\:{with}\:{a}>\mathrm{0}\:\:{L}\:{means}\:{laplace}\:{transform}\:. \\ $$

Question Number 42488 Answers: 1 Comments: 1

find ∫ (1+(1/t^2 ))arctan(t−(1/t))dt .

$${find}\:\:\int\:\:\:\left(\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right){arctan}\left({t}−\frac{\mathrm{1}}{{t}}\right){dt}\:. \\ $$

Question Number 42487 Answers: 0 Comments: 3

let f(x) = ∫_0 ^1 (dt/(x +ch(t))) 1) find a explicite form of f(x) 2) calculate ∫_0 ^1 (dt/((x+ch(t))^2 )) 3) find the value of ∫_0 ^1 (dt/(1+ch(t))) and ∫_0 ^1 (dt/(2+ch(t))) 4) find the value of ∫_0 ^1 (dt/((1+cht)^2 )) and ∫_0 ^1 (dt/((2+cht)^2 ))

$${let}\:\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\:\frac{{dt}}{{x}\:+{ch}\left({t}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\left({x}+{ch}\left({t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{ch}\left({t}\right)}\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\mathrm{2}+{ch}\left({t}\right)} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{cht}\right)^{\mathrm{2}} }\:\:{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\left(\mathrm{2}+{cht}\right)^{\mathrm{2}} } \\ $$

Question Number 42479 Answers: 1 Comments: 0

two digit number is seven times the sum of its digits.if 27 is substracted from the number its digits get interchanged. find the number

Question Number 42481 Answers: 0 Comments: 0

find f(x)= ∫_0 ^(π/4) ln(1+xtant)dt .

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}+{xtant}\right){dt}\:. \\ $$

Question Number 42482 Answers: 0 Comments: 3

let f(x)=e^(−2x) arctan(x) 1) calculate f^((n)) (x) 2) calculate f^((n)) (0) 3) developp f at integr serie .

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

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