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AllQuestion and Answers: Page 1519

Question Number 56404 Answers: 1 Comments: 0

Question Number 56392 Answers: 2 Comments: 2

find the integral of ∫x(√(x−1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{of}\:\:\int\mathrm{x}\sqrt{\mathrm{x}−\mathrm{1}}\mathrm{dx} \\ $$

Question Number 56401 Answers: 2 Comments: 1

∫((x+sin x)/(1+cos x))dx

$$\int\frac{{x}+\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{cos}\:{x}}{dx} \\ $$

Question Number 56390 Answers: 1 Comments: 2

Question Number 56386 Answers: 0 Comments: 0

Question Number 56384 Answers: 1 Comments: 2

Question Number 56383 Answers: 2 Comments: 4

Question Number 56378 Answers: 0 Comments: 2

Question Number 56368 Answers: 1 Comments: 2

What is the common formula to obtain the 3 solutions of a polynomial equation in the following form ? ax^3 + b^2 x + cx + d = 0

$${What}\:{is}\:{the}\:{common}\:{formula}\:{to}\:{obtain} \\ $$$${the}\:\mathrm{3}\:{solutions}\:{of}\:{a}\:{polynomial}\:{equation} \\ $$$${in}\:{the}\:{following}\:{form}\:? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{3}} \:+\:{b}^{\mathrm{2}} {x}\:+\:{cx}\:+\:{d}\:=\:\mathrm{0} \\ $$

Question Number 56358 Answers: 0 Comments: 3

∫_0 ^∞ (cot^(−1) x)^2 dx

$$\int_{\mathrm{0}} ^{\infty} \left(\mathrm{cot}^{−\mathrm{1}} {x}\right)^{\mathrm{2}} {dx} \\ $$$$ \\ $$

Question Number 56356 Answers: 3 Comments: 1

Find the minimum value of the function F(x) = log_e x − x for x > 0. Hence show that: log_e x ≤ x − 1. For all x > 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{F}\left(\mathrm{x}\right)\:=\:\:\mathrm{log}_{\mathrm{e}} \mathrm{x}\:\:−\:\:\mathrm{x}\:\:\:\mathrm{for}\:\:\:\mathrm{x}\:>\:\mathrm{0}. \\ $$$$\:\:\mathrm{Hence}\:\mathrm{show}\:\mathrm{that}:\:\:\:\mathrm{log}_{\mathrm{e}} \mathrm{x}\:\:\leqslant\:\:\mathrm{x}\:−\:\mathrm{1}.\:\:\mathrm{For}\:\mathrm{all}\:\:\:\mathrm{x}\:>\:\mathrm{0} \\ $$

Question Number 56351 Answers: 1 Comments: 0

log_3 (2y+1)+2=log_3 6y^2 −log_3 2y

$$\mathrm{log}_{\mathrm{3}} \left(\mathrm{2y}+\mathrm{1}\right)+\mathrm{2}=\mathrm{log}_{\mathrm{3}} \mathrm{6y}^{\mathrm{2}} −\mathrm{log}_{\mathrm{3}} \mathrm{2y} \\ $$

Question Number 56339 Answers: 1 Comments: 0

Question Number 56336 Answers: 1 Comments: 0

make r the subject of P=(1+(r/(100)))^t

$$\mathrm{make}\:\:\mathrm{r}\:\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\:\: \\ $$$$\mathrm{P}=\left(\mathrm{1}+\frac{\mathrm{r}}{\mathrm{100}}\right)^{\mathrm{t}} \\ $$

Question Number 56335 Answers: 2 Comments: 0

If b^x = ((b/k))^y = k^m and b ≠ 1 show that: (1/x) = (1/y) = (1/m)

$$\mathrm{If}\:\:\:\:\:\:\mathrm{b}^{\mathrm{x}} \:\:=\:\:\left(\frac{\mathrm{b}}{\mathrm{k}}\right)^{\mathrm{y}} \:\:=\:\:\mathrm{k}^{\mathrm{m}} \:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\mathrm{b}\:\neq\:\mathrm{1} \\ $$$$\mathrm{show}\:\mathrm{that}:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:=\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:\frac{\mathrm{1}}{\mathrm{m}} \\ $$

Question Number 56332 Answers: 1 Comments: 0

calculate lim_(x→0) ((arctan(1+cosx)−arctan(2+x))/x^3 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{arctan}\left(\mathrm{1}+{cosx}\right)−{arctan}\left(\mathrm{2}+{x}\right)}{{x}^{\mathrm{3}} } \\ $$

Question Number 56331 Answers: 0 Comments: 0

calculate lim_(x→0) ((ch(cosx)−cos(chx))/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{ch}\left({cosx}\right)−{cos}\left({chx}\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 56330 Answers: 0 Comments: 1

calculate Σ_(n=0) ^∞ (((−1)^n )/(n^2 (3n+1)))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left(\mathrm{3}{n}+\mathrm{1}\right)} \\ $$

Question Number 56329 Answers: 0 Comments: 1

1)calculate A_n =∫_(1/n) ^1 ((ln(1+x^2 ))/(1+x^2 ))dx with n integr and n≥1 2) find lim_(n→+∞) A_n 3) study the convergence of Σ A_n

$$\left.\mathrm{1}\right){calculate}\:{A}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{A}_{{n}} \\ $$

Question Number 56345 Answers: 0 Comments: 1

let f(a) =∫_0 ^∞ (dx/(x^n +a^n )) with n integr ≥2 and a>0 1) calculate f(a) intems of a 2) let g(a) =∫_0 ^∞ (dx/((x^n +a^n )^2 )) calculate g(a) interms of a 3) find the values of integrals ∫_0 ^∞ (dx/(x^8 +16)) and ∫_0 ^∞ (dx/((x^8 +16)^2 ))

$${let}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{{x}^{{n}} \:+{a}^{{n}} }\:\:\:{with}\:{n}\:{integr}\:\geqslant\mathrm{2}\:\:{and}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{intems}\:{of}\:{a} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{{n}} \:+{a}^{{n}} \right)^{\mathrm{2}} }\:\:{calculate}\:{g}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{8}} +\mathrm{16}}\:\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{8}} \:+\mathrm{16}\right)^{\mathrm{2}} } \\ $$

Question Number 56500 Answers: 1 Comments: 1

find the X_0 value of the x variable with which the function assumes the lower value. f(x)=x^2 −3x+7+sin(πx)

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{X}}_{\mathrm{0}} \:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{x}}\:\boldsymbol{{variable}}\:\boldsymbol{{with}}\:\boldsymbol{{which}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{function}}\:\boldsymbol{{assumes}}\:\boldsymbol{{the}}\:\boldsymbol{{lower}}\:\boldsymbol{{value}}. \\ $$$$ \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{7}+{sin}\left(\pi{x}\right) \\ $$

Question Number 56503 Answers: 1 Comments: 0

A boat travels 30km upstream and 44km downstream in 10 hours. in 13 hours it can travel 40km upstream and 55km downstream. Determine the speed of the stream and that of the boat in still water. (in km/hr)

$$\mathrm{A}\:\mathrm{boat}\:\mathrm{travels}\:\mathrm{30km}\:\mathrm{upstream}\:\mathrm{and}\: \\ $$$$\mathrm{44km}\:\mathrm{downstream}\:\mathrm{in}\:\mathrm{10}\:\mathrm{hours}.\: \\ $$$$\mathrm{in}\:\mathrm{13}\:\mathrm{hours}\:\mathrm{it}\:\mathrm{can}\:\mathrm{travel}\:\mathrm{40km}\:\mathrm{upstream} \\ $$$$\mathrm{and}\:\mathrm{55km}\:\mathrm{downstream}.\:\mathrm{Determine}\:\mathrm{the} \\ $$$$\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{stream}\:\:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{boat}\:\mathrm{in}\:\mathrm{still}\:\mathrm{water}.\:\left(\mathrm{in}\:\mathrm{km}/\mathrm{hr}\right) \\ $$

Question Number 56325 Answers: 0 Comments: 3

Happy π−day

$${Happy}\:\pi−{day} \\ $$

Question Number 56311 Answers: 0 Comments: 1

let f(x) =∫_0 ^∞ ((cos(xt))/(x^2 +t^2 )) dt with x>0 1) find f(x) 2) find the values of ∫_0 ^∞ ((cos(t))/(1+t^2 ))dt and ∫_0 ^∞ ((cos(2t))/(4+t^2 ))dt 3) let U_n =∫_0 ^∞ ((cos(nt))/(n^2 +t^2 ))dt find lim_(n→+∞) U_n and study the convergenge of Σ U_n and Σ U_n ^2

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\mathrm{2}{t}\right)}{\mathrm{4}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({nt}\right)}{{n}^{\mathrm{2}} +{t}^{\mathrm{2}} }{dt}\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \:\:\:\:{and}\:{study}\:{the}\:{convergenge}\:{of} \\ $$$$\Sigma\:{U}_{{n}} \:\:\:{and}\:\Sigma\:{U}_{{n}} ^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 56310 Answers: 0 Comments: 2

let f(x)=∫_(−∞) ^(+∞) cos(t^2 +xt +3)dt with x>0 1) find f(x) 2) calculate ∫_1 ^4 f(x)dx and ∫_1 ^(+∞) f(x)dx

$${let}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\:{cos}\left({t}^{\mathrm{2}} \:+{xt}\:+\mathrm{3}\right){dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{4}} {f}\left({x}\right){dx}\:{and}\:\int_{\mathrm{1}} ^{+\infty} {f}\left({x}\right){dx} \\ $$

Question Number 56321 Answers: 2 Comments: 0

Solve for x and y x (√x) + y(√y) = 182 ..... (i) x (√y) + y(√x) = 183 ..... (ii)

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\:\mathrm{x}\:\sqrt{\mathrm{x}}\:\:+\:\mathrm{y}\sqrt{\mathrm{y}}\:\:=\:\mathrm{182}\:\:\:\:\:\:.....\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\mathrm{x}\:\sqrt{\mathrm{y}}\:\:+\:\mathrm{y}\sqrt{\mathrm{x}}\:\:=\:\mathrm{183}\:\:\:\:\:\:.....\:\left(\mathrm{ii}\right) \\ $$

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