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find f(a) =∫_0 ^∞ (dx/(x^3 +a^3 )) with a>0 2)find g(a)=∫_0 ^∞ (dx/((x^3 +a^3 )^2 )) 3)find the value of ∫_0 ^∞ (dx/((1+x^3 )^2 )) 4)find the value of ∫_0 ^∞ (dx/(8x^3 +1))

$${find}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{3}} \:+{a}^{\mathrm{3}} }\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){find}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{3}} \:+{a}^{\mathrm{3}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{8}{x}^{\mathrm{3}} \:+\mathrm{1}} \\ $$

Question Number 44201 Answers: 1 Comments: 4

let f(x)=∫_0 ^1 ((ln(1+xt^2 ))/t^2 )dt with x ∈R 1) find a explicit form of f(x) 2)calculate ∫_0 ^1 ((ln(1+t^2 ))/t^2 )dt 3)calculate ∫_0 ^1 ((ln(1+2t^2 ))/t^2 )dt 4) calculate ∫_0 ^1 ((ln(1−t^2 ))/t^2 )dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}\:\in{R} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 44179 Answers: 1 Comments: 1

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

$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} −{x}\:+\mathrm{1}}} \\ $$

Question Number 44178 Answers: 0 Comments: 0

find f(x)=∫_0 ^π ((sin^2 t)/((x^2 −2x cost +1)^2 ))dt 2)find the value of ∫_0 ^π ((sin^2 t)/((x^2 −cost +1)^2 ))dt

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} {t}}{\left({x}^{\mathrm{2}} −\mathrm{2}{x}\:{cost}\:+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} {t}}{\left({x}^{\mathrm{2}} −{cost}\:+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 44176 Answers: 0 Comments: 1

find A_n =∫_0 ^∞ (t^n −[t])e^(−nt) dt and lim_(n→+∞) A_n n integr natural.

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \left({t}^{{n}} −\left[{t}\right]\right){e}^{−{nt}} {dt} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$${n}\:{integr}\:{natural}. \\ $$

Question Number 44175 Answers: 0 Comments: 2

find f(a) =∫_1 ^(+∞) (dx/(ch^2 x +a sh^2 x)) 2) find the value of ∫_1 ^(+∞) (dx/(ch^2 x+2sh^2 x))