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Question Number 66071    Answers: 1   Comments: 0

Evaluate the integral as a limit of sums: 1.∫_1 ^3 (e^(2−3x) +x^2 +1)dx

$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{as}\:\mathrm{a}\:\mathrm{limit}\:\mathrm{of}\:\mathrm{sums}: \\ $$$$\mathrm{1}.\int_{\mathrm{1}} ^{\mathrm{3}} \left({e}^{\mathrm{2}−\mathrm{3}{x}} +{x}^{\mathrm{2}} +\mathrm{1}\right){dx} \\ $$

Question Number 66066    Answers: 0   Comments: 0

Integrate ∫(dx/(ax^2 +bx+c)) y = ax^2 +bx+c y′=(dy/dx)=2ax+b d = (√(b^2 −4ac)) d′ = (√(−d)) Case 1. d^2 < 0 I=(2/d′)tan^(−1) ((y′)/d′)+C [tan^(−1) α=arctan α] Case 2. d^2 =0 I=((−2)/(y′))+C Case 3. d^2 > 0 I=(1/d)ln∣((y′−d)/(y′+d))∣+C [ln a = log_e a]

$${Integrate}\:\int\frac{{dx}}{{ax}^{\mathrm{2}} +{bx}+{c}} \\ $$$${y}\:=\:{ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${y}'=\frac{{dy}}{{dx}}=\mathrm{2}{ax}+{b} \\ $$$${d}\:=\:\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}} \\ $$$${d}'\:=\:\sqrt{−{d}} \\ $$$${Case}\:\mathrm{1}.\:{d}^{\mathrm{2}} \:<\:\mathrm{0} \\ $$$${I}=\frac{\mathrm{2}}{{d}'}{tan}^{−\mathrm{1}} \frac{{y}'}{{d}'}+{C}\:\:\:\left[{tan}^{−\mathrm{1}} \alpha={arctan}\:\alpha\right] \\ $$$${Case}\:\mathrm{2}.\:{d}^{\mathrm{2}} =\mathrm{0} \\ $$$${I}=\frac{−\mathrm{2}}{{y}'}+{C} \\ $$$${Case}\:\mathrm{3}.\:{d}^{\mathrm{2}} \:>\:\mathrm{0} \\ $$$${I}=\frac{\mathrm{1}}{{d}}\mathrm{ln}\mid\frac{{y}'−{d}}{{y}'+{d}}\mid+{C}\:\:\:\left[\mathrm{ln}\:{a}\:=\:\mathrm{log}_{{e}} \:{a}\right] \\ $$

Question Number 66065    Answers: 0   Comments: 2

find the value of U_n =∫_(−∞) ^(+∞) e^(−nx^2 ) sin(x^2 −2x)dx find nature of the serie Σ U_n and Σe^(−n^2 ) U_n

$${find}\:{the}\:{value}\:{of}\:{U}_{{n}} =\int_{−\infty} ^{+\infty} {e}^{−{nx}^{\mathrm{2}} } {sin}\left({x}^{\mathrm{2}} −\mathrm{2}{x}\right){dx} \\ $$$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \:{and}\:\Sigma{e}^{−{n}^{\mathrm{2}} } {U}_{{n}} \\ $$

Question Number 66064    Answers: 0   Comments: 1

find the value of ∫_(−∞) ^(+∞) cos(x^2 −x+1)dx

$${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{+\infty} \:{cos}\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right){dx} \\ $$

Question Number 66063    Answers: 0   Comments: 0

let x^2 −x +lnx =0 by using newton method find a approximate value of the roots of this equation.

$${let}\:\:\:{x}^{\mathrm{2}} −{x}\:+{lnx}\:=\mathrm{0}\:\:\:\:\:{by}\:{using}\:{newton}\:{method}\:{find} \\ $$$${a}\:{approximate}\:{value}\:{of}\:{the}\:{roots}\:{of}\:{this}\:{equation}. \\ $$$$ \\ $$

Question Number 66062    Answers: 0   Comments: 3

let f(x) =∫_0 ^1 (dt/(ch(t)+xsh(t))) 1) find a explicit form of f(x) 2) determine g(x) =∫_0 ^1 (dt/((ch(t)+xsh(t))^2 )) 3) calculate ∫_0 ^1 (dt/(ch(t)+3sh(t))) and ∫_0 ^1 (dt/({ch(t)+3sh(t)}^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{ch}\left({t}\right)+{xsh}\left({t}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left({ch}\left({t}\right)+{xsh}\left({t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{ch}\left({t}\right)+\mathrm{3}{sh}\left({t}\right)}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left\{{ch}\left({t}\right)+\mathrm{3}{sh}\left({t}\right)\right\}^{\mathrm{2}} } \\ $$

Question Number 66061    Answers: 0   Comments: 0

let f(t) =∫_0 ^1 ((sinx)/(1+te^(−x^2 ) ))dx with ∣t∣<1 developp f at integr serie .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{sinx}}{\mathrm{1}+{te}^{−{x}^{\mathrm{2}} } }{dx}\:\:\:\:{with}\:\mid{t}\mid<\mathrm{1} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 66060    Answers: 0   Comments: 3

let f(x) =∫_0 ^(π/4) (dt/(x+tant)) with x real 1) find aexplicit form of f(x) 2)find also g(x) =∫_0 ^(π/4) (dt/((x+tant)^2 )) 3)give f^((n)) (x)at form of integral 4)calculate ∫_0 ^(π/4) (dt/(2+tant)) and ∫_0 ^(π/4) (dt/((2+tant)^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{{x}+{tant}}\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{aexplicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left({x}+{tant}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right){at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right){calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\mathrm{2}+{tant}}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\left(\mathrm{2}+{tant}\right)^{\mathrm{2}} } \\ $$

Question Number 66058    Answers: 2   Comments: 5

If x + (1/x)=1 find out value:− ((x^(20) +x^(17) +x^(14) +x^(11) )/(x^(17) +x^(14) +x^(11) +x^8 )) = ?

$${If}\:\:\:\:{x}\:+\:\frac{\mathrm{1}}{{x}}=\mathrm{1} \\ $$$$ \\ $$$${find}\:{out}\:{value}:− \\ $$$$ \\ $$$$\:\:\:\:\frac{{x}^{\mathrm{20}} +{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} }{{x}^{\mathrm{17}} +{x}^{\mathrm{14}} +{x}^{\mathrm{11}} +{x}^{\mathrm{8}} }\:\:\:\:=\:\:\:\:? \\ $$

Question Number 66055    Answers: 1   Comments: 1

Question Number 66048    Answers: 0   Comments: 1

∫(x/(√(ln(1/x)))) dx

$$\int\frac{{x}}{\sqrt{{ln}\left(\mathrm{1}/{x}\right)}}\:{dx} \\ $$

Question Number 66046    Answers: 0   Comments: 2

find (dy/dx) if y=x^x^x help pls

$${find}\:\frac{{dy}}{{dx}}\:{if}\:{y}={x}^{{x}^{{x}} } \\ $$$${help}\:{pls} \\ $$

Question Number 66044    Answers: 1   Comments: 0

Question Number 66036    Answers: 1   Comments: 0

Question Number 66032    Answers: 0   Comments: 2

simplify w_n =(1+in)^n −(1−in)^n with n integr natural

$${simplify}\:\:{w}_{{n}} =\left(\mathrm{1}+{in}\right)^{{n}} −\left(\mathrm{1}−{in}\right)^{{n}} \:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 66019    Answers: 2   Comments: 1

lim_(x→∞) (1 + (2/x))^x =

$$\underset{{x}\rightarrow\infty} {{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{2}}{{x}}\right)^{{x}} \:= \\ $$

Question Number 66018    Answers: 1   Comments: 1

prove by mathematical induction that Σ_(r=1) ^n r(r + 1) = (n/3)(n + 1)(n + 2)

$${prove}\:{by}\:{mathematical}\:{induction}\:{that}\: \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left({n}\:+\:\mathrm{2}\right) \\ $$

Question Number 66017    Answers: 0   Comments: 0

show that the equation xe^x =1 has a root between 0.5 and 0.6 starting with 0.55 as a first approximate.

$${show}\:{that}\:{the}\:{equation}\:{xe}^{{x}} =\mathrm{1}\:{has}\:{a}\:{root}\:{between}\:\mathrm{0}.\mathrm{5}\:{and}\:\mathrm{0}.\mathrm{6}\:{starting} \\ $$$${with}\:\mathrm{0}.\mathrm{55}\:{as}\:{a}\:{first}\:{approximate}. \\ $$

Question Number 66016    Answers: 1   Comments: 6

Evaluate a. ∫_1 ^2 (lnx)^2 dx b. ∫_0 ^(π/6) sin^2 x cos^3 xdx

$${Evaluate}\:\:\: \\ $$$${a}.\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\left({lnx}\right)^{\mathrm{2}} {dx} \\ $$$${b}.\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:{sin}^{\mathrm{2}} {x}\:{cos}^{\mathrm{3}} {xdx} \\ $$

Question Number 66010    Answers: 0   Comments: 1

Using Σ_(n=1) ^∞ (1/(n!))=e , prove that lim_(n→∞) (1+(1/n))^n =e

$${Using}\:\sum_{{n}=\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{{n}!}={e}\:,\:{prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} ={e} \\ $$

Question Number 66005    Answers: 0   Comments: 0

solve x^2 y^(′′) +xy^′ −3y =4e^(−x)

$${solve}\:\:\:{x}^{\mathrm{2}} {y}^{''} \:+{xy}^{'} −\mathrm{3}{y}\:=\mathrm{4}{e}^{−{x}} \\ $$

Question Number 66004    Answers: 1   Comments: 0

x^x^x^⋰ =3 find the value of x

$${x}^{{x}^{{x}^{\iddots} } } =\mathrm{3} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x} \\ $$

Question Number 65992    Answers: 1   Comments: 1

x = (1/(3∙1!)) + (1/(4∙2!)) + (1/(5∙3!)) + … + (1/(1002∙1000!)) x∙1000! = ?

$${x}\:\:=\:\:\frac{\mathrm{1}}{\mathrm{3}\centerdot\mathrm{1}!}\:+\:\frac{\mathrm{1}}{\mathrm{4}\centerdot\mathrm{2}!}\:+\:\frac{\mathrm{1}}{\mathrm{5}\centerdot\mathrm{3}!}\:+\:\ldots\:+\:\frac{\mathrm{1}}{\mathrm{1002}\centerdot\mathrm{1000}!} \\ $$$${x}\centerdot\mathrm{1000}!\:\:=\:\:? \\ $$

Question Number 65988    Answers: 3   Comments: 1

∫dx/x^2 −x+1

$$\int{dx}/{x}^{\mathrm{2}} −{x}+\mathrm{1} \\ $$

Question Number 66041    Answers: 0   Comments: 0

Question Number 65983    Answers: 1   Comments: 1

Simplify (1+ 2i(√2))^7 − (1 +2i)^7

$$\:{Simplify}\: \\ $$$$\:\:\:\left(\mathrm{1}+\:\mathrm{2}{i}\sqrt{\mathrm{2}}\right)^{\mathrm{7}} \:−\:\left(\mathrm{1}\:+\mathrm{2}{i}\right)^{\mathrm{7}} \\ $$

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