Question and Answers Forum

let x>0 and f(x)=∫_1 ^2 (t+1)(√(t^2 −2xt−1))dt 1) find a explicit form of f(x) 2) determine also g(x)=∫_1 ^2 ((t^2 +t)/(√(t^2 −2xt−1)))dt 3)find the value of integrals ∫_1 ^2 (t+1)(√(t^2 −t−1))dt and ∫_1 ^2 ((t^(2 ) +t)/(√(t^2 −t−1)))dt .

$${let}\:{x}>\mathrm{0}\:{and}\:{f}\left({x}\right)=\int_{\mathrm{1}} ^{\mathrm{2}} \left({t}+\mathrm{1}\right)\sqrt{{t}^{\mathrm{2}} −\mathrm{2}{xt}−\mathrm{1}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({x}\right)=\int_{\mathrm{1}} ^{\mathrm{2}} \frac{{t}^{\mathrm{2}} \:+{t}}{\sqrt{{t}^{\mathrm{2}} −\mathrm{2}{xt}−\mathrm{1}}}{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{integrals}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \left({t}+\mathrm{1}\right)\sqrt{{t}^{\mathrm{2}} −{t}−\mathrm{1}}{dt} \\ $$$${and}\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\frac{{t}^{\mathrm{2}\:} +{t}}{\sqrt{{t}^{\mathrm{2}} −{t}−\mathrm{1}}}{dt}\:. \\ $$$$ \\ $$

Question Number 66213 Answers: 0 Comments: 3

calculate A_n =∫_0 ^∞ cos(x^n )dx and B_n =∫_0 ^∞ sin(x^n )dx with n integr and n≥2

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{{n}} \right){dx}\:{and}\:{B}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{{n}} \right){dx} \\ $$$${with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$

Question Number 66194 Answers: 0 Comments: 0

∫(x^a /(bx^n +c)) dx

$$\int\frac{{x}^{{a}} }{{bx}^{{n}} +{c}}\:{dx} \\ $$

Question Number 66170 Answers: 0 Comments: 1

calculate ∫_0 ^∞ sin(x^3 )dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{sin}\left({x}^{\mathrm{3}} \right){dx} \\ $$

Question Number 66169 Answers: 0 Comments: 1

find the values of ∫_0 ^∞ cos(x^2 )dx and ∫_0 ^∞ sin(x^2 )dx(fresnel integrals) by using Γ(z) =∫_0 ^∞ t^(z−1) e^(−t) dt

$${find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{\mathrm{2}} \right){dx}\left({fresnel}\:{integrals}\right) \\ $$$${by}\:{using}\:\Gamma\left({z}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{z}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:\: \\ $$

Question Number 66168 Answers: 0 Comments: 0

prove without calculus that ∫_0 ^∞ cos(x^2 )dx=∫_0 ^∞ sin(x^2 )dx

$${prove}\:{without}\:{calculus}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}=\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 66150 Answers: 0 Comments: 1

find ∫_0 ^∞ e^(−x^3 ) sin(x^3 )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{3}} } {sin}\left({x}^{\mathrm{3}} \right){dx}\: \\ $$

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 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 66082 Answers: 0 Comments: 3

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 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 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 66048 Answers: 0 Comments: 1

∫(x/(√(ln(1/x)))) dx

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

Question Number 66044 Answers: 1 Comments: 0

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 65950 Answers: 4 Comments: 0

∫_0 ^( π/2) tan^3 xdx = ?

$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{tan}\:^{\mathrm{3}} {xdx}\:=\:? \\ $$

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