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IntegrationQuestion and Answers: Page 297

Question Number 32043 Answers: 0 Comments: 0

let f(x)= ∫_x ^x^2 (dt/(lnt)) with x>0 and x≠1 1) prove that ∀ x>1 ∫_x ^x^2 ((xdt)/(tlnt)) ≤f(x)≤ ∫_x ^x^2 ((x^2 dt)/(tlnt)) after find lim_(x→1) f(x) 2) calculate f^′ (x) .

$${let}\:{f}\left({x}\right)=\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{dt}}{{lnt}}\:\:{with}\:{x}>\mathrm{0}\:{and}\:{x}\neq\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{x}>\mathrm{1}\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\frac{{xdt}}{{tlnt}}\:\leqslant{f}\left({x}\right)\leqslant\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{x}^{\mathrm{2}} {dt}}{{tlnt}}\:\:{after} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:. \\ $$

Question Number 32040 Answers: 0 Comments: 2

let give f(x) =∫_0 ^(π/2) (dt/(1+x tant)) 1) find a simple form of f(x) 2) calculate ∫_0 ^(π/2) ((tant)/((1+xtant)^2 ))dt 3)give the value of ∫_0 ^(π/2) ((tant)/((1+(√3) tant)^2 )) dt .

$${let}\:{give}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{x}\:{tant}}\:\: \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{tant}}{\left(\mathrm{1}+{xtant}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){give}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tant}}{\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{tant}\right)^{\mathrm{2}} }\:{dt}\:. \\ $$

Question Number 32039 Answers: 0 Comments: 3

a>−1 calculate ∫_0 ^(π/2) (dt/(1+a tan^2 t)) . 2) find ∫_0 ^(π/2) ((tan^2 t)/((1+atan^2 t)^2 )) dt 3) find the value of ∫_0 ^(π/2) ((tan^2 t)/((1+2tan^2 t)^2 ))dt.

$${a}>−\mathrm{1}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dt}}{\mathrm{1}+{a}\:{tan}^{\mathrm{2}} {t}}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+{atan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{3}\right)\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{tan}^{\mathrm{2}} {t}}{\left(\mathrm{1}+\mathrm{2}{tan}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }{dt}.\: \\ $$

Question Number 32034 Answers: 0 Comments: 0

let u_n = ∫_0 ^1 (dx/(1+x+...+x^n )) study the convergence of Σ u_n .

$${let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+...+{x}^{{n}} }\:\:{study}\:{the}\:{convergence}\:{of} \\ $$$$\Sigma\:{u}_{{n}} \:\:. \\ $$

Question Number 32031 Answers: 0 Comments: 1

let f(a) = ∫_0 ^∞ e^(−ax) ln(x)dx with a>0 1) find f(a) 2) find ∫_0 ^∞ e^(−ax) (xlnx)dx 3) calculate ∫_0 ^∞ e^(−2x) (xlnx)dx .

$${let}\:\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} {ln}\left({x}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{ax}} \left({xlnx}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}} \left({xlnx}\right){dx}\:\:. \\ $$

Question Number 32029 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−αx) ln(x) dx with α>0 .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\alpha{x}} {ln}\left({x}\right)\:{dx}\:\:{with}\:\:\alpha>\mathrm{0}\:. \\ $$

Question Number 32026 Answers: 0 Comments: 1

let α>0 prove that Σ_(n=0) ^∞ (((−1)^n )/(n+α)) =∫_0 ^1 (x^(α−1) /(1+x))dx .

$${let}\:\alpha>\mathrm{0}\:{prove}\:{that}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\alpha}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\alpha−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:. \\ $$

Question Number 31974 Answers: 0 Comments: 0

1)find I(p,q) = ∫_0 ^1 t^p (1−t)^q dt with pand q integrs 2) find the nature of Σ I_((n,n))

$$\left.\mathrm{1}\right){find}\:{I}\left({p},{q}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{t}^{{p}} \:\left(\mathrm{1}−{t}\right)^{{q}} \:{dt}\:\:{with}\:{pand}\:{q}\:{integrs} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:\:{I}_{\left({n},{n}\right)} \\ $$

Question Number 31970 Answers: 0 Comments: 0

find the nature of ∫_2 ^∞ (e^(−x) /(√(x^2 −4))) dx .

$${find}\:{the}\:{nature}\:{of}\:\:\int_{\mathrm{2}} ^{\infty} \:\:\frac{{e}^{−{x}} }{\sqrt{{x}^{\mathrm{2}} \:−\mathrm{4}}}\:{dx}\:. \\ $$

Question Number 31969 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (((1+t^2 )/(1+t^4 )))arctant dt.

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\left(\frac{\mathrm{1}+{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{4}} }\right){arctant}\:{dt}. \\ $$

Question Number 31968 Answers: 0 Comments: 0

find ∫_2 ^(√5) x(√((x−2)((√5)−x))) dx .

$${find}\:\:\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} {x}\sqrt{\left({x}−\mathrm{2}\right)\left(\sqrt{\mathrm{5}}−{x}\right)}\:{dx}\:. \\ $$

Question Number 31967 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((arctanx)/(x^2 +x+1))dx .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctanx}}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}{dx}\:. \\ $$

Question Number 31951 Answers: 1 Comments: 0

Evaluate ∫sin (√x)dx

$${Evaluate}\:\int\mathrm{sin}\:\sqrt{{x}}{dx} \\ $$

Question Number 31858 Answers: 0 Comments: 1

∫((sinx)/x)dx

$$\int\frac{{sinx}}{{x}}{dx} \\ $$

Question Number 31839 Answers: 0 Comments: 1

I = ∫ (√(x + (√(x^2 − 1)))) dx

$${I}\:=\:\int\:\sqrt{{x}\:+\:\sqrt{{x}^{\mathrm{2}} \:−\:\mathrm{1}}}\:{dx} \\ $$

Question Number 31838 Answers: 0 Comments: 0

Given f(x) = (3/(16)) (∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Solve lim_(t→0) ((2t + (∫_(f(2) + 2) ^(f^(−1) (t)) [f ′(x)]^2 dx))/(1 − cos t cosh 2t cos 3t))

$$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} \:{f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} \:{f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Solve} \\ $$$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{t}\:+\:\left(\int_{{f}\left(\mathrm{2}\right)\:+\:\mathrm{2}} ^{{f}^{−\mathrm{1}} \left({t}\right)} \left[{f}\:'\left({x}\right)\right]^{\mathrm{2}} \:{dx}\right)}{\mathrm{1}\:−\:\mathrm{cos}\:{t}\:\mathrm{cosh}\:\mathrm{2}{t}\:\mathrm{cos}\:\mathrm{3}{t}} \\ $$

Question Number 31787 Answers: 2 Comments: 0

∫((4x−3)/(x^2 +3x+8))dx

$$\int\frac{\mathrm{4}{x}−\mathrm{3}}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{8}}{dx} \\ $$

Question Number 31747 Answers: 0 Comments: 1

let give ∣λ∣<1 and u_n = ∫_0 ^π ((cos(nx))/(1−2λ cosx +λ^2 )) prove that Σ_(n=0) ^∞ u_n is convergent and find its sum .

$${let}\:{give}\:\mid\lambda\mid<\mathrm{1}\:{and}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{cos}\left({nx}\right)}{\mathrm{1}−\mathrm{2}\lambda\:{cosx}\:+\lambda^{\mathrm{2}} } \\ $$$${prove}\:{that}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{sum}\:. \\ $$

Question Number 31517 Answers: 0 Comments: 1

find ∫_(−1) ^1 (dx/((√(1+x)) +(√(1−x)))) .

$${find}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:\:. \\ $$

Question Number 31516 Answers: 1 Comments: 1

find ∫ (dx/(x +(√(1+x^2 )))) .

$${find}\:\int\:\:\:\frac{{dx}}{{x}\:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 31515 Answers: 1 Comments: 1

calculate ∫_0 ^1 (dx/(chx)) .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{{chx}}\:. \\ $$

Question Number 31514 Answers: 1 Comments: 0

find ∫_0 ^1 ((arctan(2x))/((1+x)^2 ))dx.

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

Question Number 31513 Answers: 1 Comments: 1

find ∫_0 ^(2π) (dx/(2 +cosx)) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dx}}{\mathrm{2}\:+{cosx}}\:\:. \\ $$

Question Number 31512 Answers: 0 Comments: 1

find lim_(x→∞) ∫_x ^(2x) ((cos((1/t)))/t) dt.

$${find}\:{lim}_{{x}\rightarrow\infty} \:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}\:{dt}. \\ $$

Question Number 31507 Answers: 0 Comments: 0

g is real function continue let f(x)=∫_0 ^x sin(x−t)g(t)dt 1)prove that f^′ (x)= ∫_0 ^x cos(t−x)g(t)dt 2)prove that f is so<ution of the diff.equa. y^(′′) +y =g(x)

$${g}\:{is}\:{real}\:{function}\:{continue}\:{let} \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{sin}\left({x}−{t}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}^{'} \left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {cos}\left({t}−{x}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{so}<{ution}\:{of}\:{the}\:{diff}.{equa}. \\ $$$${y}^{''} \:+{y}\:={g}\left({x}\right) \\ $$

Question Number 31506 Answers: 0 Comments: 1

let f(x)=∫_x ^(2x) ((sht)/t)dt 1) calculate f^′ (x) 2) find lim_(x→0) f(x) .

$${let}\:{f}\left({x}\right)=\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{sht}}{{t}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{f}\left({x}\right)\:. \\ $$

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