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Question Number 66098 Answers: 0 Comments: 1

Question Number 66096 Answers: 0 Comments: 2

∫ ((√((1+x^2 )))/x^2 ) dx = ?

$$\int\:\:\frac{\sqrt{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 66090 Answers: 1 Comments: 0

∫_( 0) ^(π/2) ((sin^2 x)/(sin x+cos x)) dx =

$$\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}^{\mathrm{2}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:{dx}\:= \\ $$

Question Number 66089 Answers: 0 Comments: 0

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Question Number 66085 Answers: 0 Comments: 0

calculate ∫_0 ^1 ((arctan((√(x^2 +2))))/((x^2 +1)(√(x^2 +2))))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}}}{dx} \\ $$

Question Number 66084 Answers: 1 Comments: 1

How to solve this limit? lim_(x→∞) (7x+(2/x))^x

$${How}\:{to}\:{solve}\:{this}\:{limit}? \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\left(\mathrm{7}{x}+\frac{\mathrm{2}}{{x}}\right)^{{x}} \\ $$

Question Number 66082 Answers: 0 Comments: 3

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