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Question Number 42823 Answers: 1 Comments: 1

Evaluate : ∫_(−5) ^( 5) x^2 [x+(1/2)]dx = ? where [.]= greatest integer function

$$\mathrm{Evaluate}\:: \\ $$$$\int_{−\mathrm{5}} ^{\:\mathrm{5}} \:{x}^{\mathrm{2}} \left[{x}+\frac{\mathrm{1}}{\mathrm{2}}\right]{dx}\:=\:\:? \\ $$$${where}\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$

Question Number 42812 Answers: 0 Comments: 1

study the convervence of ∫_1 ^(+∞) ((arctan(x−1))/((x^2 −1)^(4/3) )) dx

$${study}\:{the}\:{convervence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({x}−\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }\:{dx} \\ $$

Question Number 42810 Answers: 1 Comments: 1

find ∫_0 ^∞ (x^5 /(1+x^7 ))dx .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{5}} }{\mathrm{1}+{x}^{\mathrm{7}} }{dx}\:\:. \\ $$

Question Number 42809 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((tdt)/((1+t^4 )^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{tdt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$

Question Number 42808 Answers: 1 Comments: 0

let f(x) = ∫_x ^(2x) ((1+cos(t))/(√(t^4 −t^2 +4)))dt 1) find D_f 2) calculate f^′ (x) 3) find lim_(x→+∞) f(x)

$${let}\:{f}\left({x}\right)\:=\:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{\mathrm{1}+{cos}\left({t}\right)}{\sqrt{{t}^{\mathrm{4}} −{t}^{\mathrm{2}} \:+\mathrm{4}}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{lim}_{{x}\rightarrow+\infty} \:{f}\left({x}\right) \\ $$

Question Number 42806 Answers: 0 Comments: 0

let u_n = ∫_n ^(n+2) (((t+n)^(1/4) )/t^(1/3) )dt find lim_(n→+∞) u_n

$${let}\:\:{u}_{{n}} =\:\int_{{n}} ^{{n}+\mathrm{2}} \:\:\:\frac{\left({t}+{n}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{t}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 42805 Answers: 1 Comments: 1

calculate lim_(n→+∞) S_n with S_n = (1/n^4 ) Σ_(k=1) ^n (k^3 /(√((1+((k/n))^2 )^3 )))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \:\:\:{with} \\ $$$${S}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}^{\mathrm{3}} }{\sqrt{\left(\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} \right)^{\mathrm{3}} }} \\ $$

Question Number 42804 Answers: 1 Comments: 1

calculate ∫_(1/2) ^1 (dx/((√(4x^2 −1)) +(√(4x^2 +1))))

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

Question Number 42803 Answers: 1 Comments: 0

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

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

Question Number 42802 Answers: 0 Comments: 1

calculate ∫_(1/2) ^(5/4) (x^3 /(√(2+x−x^2 )))dx

$${calculate}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{5}}{\mathrm{4}}} \:\:\:\frac{{x}^{\mathrm{3}} }{\sqrt{\mathrm{2}+{x}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 42801 Answers: 1 Comments: 1

find f(x) = ∫_(π/4) ^(π/3) ((cosxdx)/(2cos^2 x +sin^2 x +1))

$${find}\:{f}\left({x}\right)\:=\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{cosxdx}}{\mathrm{2}{cos}^{\mathrm{2}} {x}\:+{sin}^{\mathrm{2}} {x}\:+\mathrm{1}} \\ $$

Question Number 42800 Answers: 1 Comments: 0

find ∫ ((sinx)/(1+2 cosx))dx

$${find}\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+\mathrm{2}\:{cosx}}{dx} \\ $$

Question Number 42799 Answers: 0 Comments: 2

let I = ∫_0 ^(π/8) e^(−2t) cos^4 t and J =∫_0 ^(π/8) e^(−2t) sin^4 dt find the values of I andJ .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:\:\:\:{and}\:{J}\:\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {dt} \\ $$$${find}\:{the}\:{values}\:{of}\:{I}\:{andJ}\:. \\ $$

Question Number 42798 Answers: 1 Comments: 2

find I_n = ∫_0 ^1 x^n (√(1−x^2 ))dx

$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 42797 Answers: 0 Comments: 1

let u_k = ∫_(−(π/2) +kπ) ^(−(π/2) +(k+1)π) e^(−t) cost dt 1) calculate u_k 2) let A_n =Σ_(k=0) ^n u_k find lim_(n→+∞) A_n

$${let}\:{u}_{{k}} =\:\int_{−\frac{\pi}{\mathrm{2}}\:+{k}\pi} ^{−\frac{\pi}{\mathrm{2}}\:+\left({k}+\mathrm{1}\right)\pi} \:\:{e}^{−{t}} \:{cost}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{u}_{{k}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{u}_{{k}} \:\:\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 42796 Answers: 1 Comments: 1

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

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

Question Number 42795 Answers: 0 Comments: 3

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

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

Question Number 42793 Answers: 1 Comments: 0

find f(a)= ∫_0 ^1 (dt/((a^2 +t^2 )^3 )) with a>0

$${find}\:{f}\left({a}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\left({a}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 42792 Answers: 1 Comments: 1

find ∫_0 ^1 ((1+x^2 )/(1+x^3 ))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{3}} }{dx}\: \\ $$

Question Number 42791 Answers: 2 Comments: 0

find ∫ (dx/((x^(2 ) +1)(√(1+x^2 ))))

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

Question Number 42790 Answers: 1 Comments: 0

1) calculate ∫_0 ^∞ (t/(1+t^4 )) dt 2) calculate ∫_0 ^1 (t/(1+t^4 ))dt 3) calculste ∫_1 ^(+∞) (t/(1+t^4 ))dt

$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculste}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\: \\ $$

Question Number 42789 Answers: 0 Comments: 1

let u_0 =1 and u_(n+1) =u_n + (2/u_n ) study the convervence of (u_n )

$${let}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{{n}+\mathrm{1}} \:={u}_{{n}} \:+\:\frac{\mathrm{2}}{{u}_{{n}} } \\ $$$${study}\:{the}\:{convervence}\:{of}\:\left({u}_{{n}} \right) \\ $$

Question Number 42788 Answers: 1 Comments: 1

calculate lim_(x→0) ((2x)/(ln(((1+x)/(1−x))))) −cosx

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{\mathrm{2}{x}}{{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)}\:−{cosx} \\ $$

Question Number 42787 Answers: 0 Comments: 0

find lim_(x→0) (1/((sinx)^4 )){ sin((x/(1−x)))−((sinx)/(1−sinx))}

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\mathrm{1}}{\left({sinx}\right)^{\mathrm{4}} }\left\{\:{sin}\left(\frac{{x}}{\mathrm{1}−{x}}\right)−\frac{{sinx}}{\mathrm{1}−{sinx}}\right\} \\ $$

Question Number 42786 Answers: 1 Comments: 1

calculate lim_(x→0) ((1−(x/(sinx)))/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{\mathrm{1}−\frac{{x}}{{sinx}}}{{x}^{\mathrm{2}} } \\ $$

Question Number 42785 Answers: 1 Comments: 1

find lim_(x→0) ((1+x −e^(arcsinx) )/x^2 )

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{\mathrm{1}+{x}\:−{e}^{{arcsinx}} }{{x}^{\mathrm{2}} } \\ $$

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