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Question Number 72430    Answers: 0   Comments: 0

Hello find finde ∫_0 ^(+∞) ((ln(x))/(x^2 +ax+b))dx conditions a^2 <4b in therm of x_1 ,x_2 root of X^2 +aX+b hint Residus theorem applied too ((log^2 (z))/(z^2 +az+b)) this is very usufull i find it in lecture yesterday because we can easly evaluat any kinde of ∫_0 ^(+∞) ((log^k (z))/(p(z)))dz withe p(z) eiwthout root in ]0,+∞[ deg(p(z))≥2

$$\mathrm{Hello}\:\mathrm{find} \\ $$$$\mathrm{finde}\:\:\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{ax}+\mathrm{b}}\mathrm{dx} \\ $$$$\mathrm{conditions}\:\mathrm{a}^{\mathrm{2}} <\mathrm{4b}\:\:\: \\ $$$$\mathrm{in}\:\mathrm{therm}\:\mathrm{of}\:\mathrm{x}_{\mathrm{1}} ,\mathrm{x}_{\mathrm{2}} \:\:\mathrm{root}\:\mathrm{of}\:\mathrm{X}^{\mathrm{2}} +\mathrm{aX}+\mathrm{b}\: \\ $$$$\:\mathrm{hint}\:\mathrm{Residus}\:\mathrm{theorem}\:\mathrm{applied}\:\mathrm{too}\:\frac{\mathrm{log}^{\mathrm{2}} \left(\mathrm{z}\right)}{\mathrm{z}^{\mathrm{2}} +\mathrm{az}+\mathrm{b}} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{very}\:\mathrm{usufull}\:\mathrm{i}\:\mathrm{find}\:\mathrm{it}\:\mathrm{in}\:\mathrm{lecture}\:\mathrm{yesterday} \\ $$$$\mathrm{because}\:\mathrm{we}\:\mathrm{can}\:\mathrm{easly}\:\mathrm{evaluat}\:\mathrm{any}\:\mathrm{kinde}\:\mathrm{of}\:\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{log}^{\mathrm{k}} \left(\mathrm{z}\right)}{\mathrm{p}\left(\mathrm{z}\right)}\mathrm{dz} \\ $$$$\left.\mathrm{withe}\:\mathrm{p}\left(\mathrm{z}\right)\:\mathrm{eiwthout}\:\:\mathrm{root}\:\mathrm{in}\:\right]\mathrm{0},+\infty\left[\:\mathrm{deg}\left(\mathrm{p}\left(\mathrm{z}\right)\right)\geqslant\mathrm{2}\right. \\ $$

Question Number 72445    Answers: 0   Comments: 0

∫((x cos(ax))/(1+x^2 )) dx

$$\int\frac{{x}\:{cos}\left({ax}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 72443    Answers: 1   Comments: 0

2[0.5{48−31}^2 +35]=?

$$\mathrm{2}\left[\mathrm{0}.\mathrm{5}\left\{\mathrm{48}−\mathrm{31}\right\}^{\mathrm{2}} +\mathrm{35}\right]=? \\ $$

Question Number 72441    Answers: 2   Comments: 1

If^n C_(12) =^n C_8 , then n=

$$\mathrm{If}\:^{{n}} {C}_{\mathrm{12}} =\:^{{n}} {C}_{\mathrm{8}} \:,\:\mathrm{then}\:{n}= \\ $$

Question Number 72439    Answers: 0   Comments: 1

Question Number 72438    Answers: 0   Comments: 0

Question Number 72421    Answers: 1   Comments: 0

Question Number 72416    Answers: 0   Comments: 1

Question Number 72414    Answers: 1   Comments: 1

Question Number 72413    Answers: 0   Comments: 0

Question Number 72408    Answers: 1   Comments: 1

Question Number 72401    Answers: 1   Comments: 0

Find the area bounded by one leaf of the rose r = 12cos (3θ).

$${Find}\:{the}\:{area}\:{bounded}\:{by}\:{one}\:{leaf}\:{of} \\ $$$${the}\:{rose}\:{r}\:=\:\mathrm{12cos}\:\left(\mathrm{3}\theta\right). \\ $$

Question Number 72398    Answers: 0   Comments: 4

lim_(x→∞) [x−x^2 ln(1+(1/x))]

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left[\mathrm{x}−\mathrm{x}^{\mathrm{2}} \mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}\right)\right] \\ $$

Question Number 72397    Answers: 0   Comments: 4

find A(x)=∫_0 ^(π/2) ln(1−xsin^2 θ)dθ with ∣x∣<1

$${find}\:{A}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−{xsin}^{\mathrm{2}} \theta\right){d}\theta\:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$

Question Number 72396    Answers: 0   Comments: 2

calculate ∫_0 ^∞ ((1+x^2 )/(2+x^2 +x^4 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{2}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 72395    Answers: 0   Comments: 0

find Σ_(k=0) ^n (C_n ^k )^3

$${find}\:\sum_{{k}=\mathrm{0}} ^{{n}} \left({C}_{{n}} ^{{k}} \right)^{\mathrm{3}} \\ $$

Question Number 72394    Answers: 0   Comments: 3

let g(x)=((ln(1+x))/(3+x^2 )) 1) find g^((n)) (x)and g^((n)) (0) 2)developp g at integr serie

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

Question Number 72393    Answers: 0   Comments: 0

let f(x) =cos(narccosx) 1)calculate f^((n)) (x) and f^((n)) (0) 2)developp f at integr serie

$${let}\:{f}\left({x}\right)\:={cos}\left({narccosx}\right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 72392    Answers: 0   Comments: 1

calculate A_n =∫_0 ^∞ e^(−nx) ln(1+x)dx with n natural ≥1

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{e}^{−{nx}} {ln}\left(\mathrm{1}+{x}\right){dx}\:\:{with}\:{n}\:{natural}\:\geqslant\mathrm{1} \\ $$

Question Number 72391    Answers: 0   Comments: 1

calculte ∫_0 ^∞ (((−1)^([x]) )/(4+x^2 ))dx

$${calculte}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{\left[{x}\right]} }{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 72362    Answers: 1   Comments: 0

Calculate the sides of a triangle knowing the heights h_(a ) =(1/9) h_b =(1/7) and h_c =(1/4)

$${Calculate}\:{the}\:{sides}\:{of}\:{a}\:{triangle} \\ $$$${knowing}\:{the}\:{heights}\:{h}_{\mathrm{a}\:} =\frac{\mathrm{1}}{\mathrm{9}} \\ $$$${h}_{\mathrm{b}} =\frac{\mathrm{1}}{\mathrm{7}}\:{and}\:{h}_{\mathrm{c}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Question Number 72462    Answers: 0   Comments: 2

lim_(x→0) ((1−(√(1+4x ))cos(x^2 ))/(x^3 arctan(x^5 )))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{4x}\:}\mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{3}} \mathrm{arctan}\left(\mathrm{x}^{\mathrm{5}} \right)} \\ $$

Question Number 72346    Answers: 0   Comments: 1

Question Number 72344    Answers: 0   Comments: 3

prove that e^(lnx) = x or a^(log_a x) = x

$${prove}\:{that}\:\:{e}^{{lnx}} \:=\:{x} \\ $$$${or}\:\:{a}^{{log}_{{a}} {x}} \:=\:{x} \\ $$

Question Number 72343    Answers: 0   Comments: 3

given that y = ln ( 1 + cos^2 x) find (dy/(dx )) at the point x = ((3π)/4) and if y =ln(x^2 + 4) find (dy/dx) at x = 1

$${given}\:{that}\:{y}\:=\:{ln}\:\left(\:\mathrm{1}\:+\:{cos}^{\mathrm{2}} {x}\right)\:{find}\:\frac{{dy}}{{dx}\:\:}\:{at}\:{the}\:{point}\:\:{x}\:=\:\frac{\mathrm{3}\pi}{\mathrm{4}} \\ $$$${and}\:\:{if}\:\:{y}\:={ln}\left({x}^{\mathrm{2}} \:+\:\mathrm{4}\right)\:{find}\:\:\frac{{dy}}{{dx}}\:{at}\:{x}\:=\:\mathrm{1} \\ $$

Question Number 72339    Answers: 0   Comments: 1

(√(x + (√(4x + (√(16x + ... + (√(4^(2019) x + 3)))))))) = (√x) + 1

$$\sqrt{{x}\:+\:\sqrt{\mathrm{4}{x}\:+\:\sqrt{\mathrm{16}{x}\:+\:...\:+\:\sqrt{\mathrm{4}^{\mathrm{2019}} {x}\:+\:\mathrm{3}}}}}\:\:=\:\:\sqrt{{x}}\:+\:\mathrm{1} \\ $$

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