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IntegrationQuestion and Answers: Page 302

Question Number 30564 Answers: 0 Comments: 0

f and g are 2 function C^n on [a,b] prove that ∫_a ^b f^((n)) (x)g(x)dx=[Σ_(k=0) ^(n−1) (−1)^k f^((k)) g^((n−k)) ]_a ^b +(−1)^n ∫_a ^b f(x)g^((n)) (x)dx

$${f}\:{and}\:{g}\:{are}\:\mathrm{2}\:{function}\:\:{C}^{{n}} \:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\int_{{a}} ^{{b}} \:{f}^{\left({n}\right)} \left({x}\right){g}\left({x}\right){dx}=\left[\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left(−\mathrm{1}\right)^{{k}} \:{f}^{\left({k}\right)} {g}^{\left({n}−{k}\right)} \right]_{{a}} ^{{b}} \:+\left(−\mathrm{1}\right)^{{n}} \int_{{a}} ^{{b}} {f}\left({x}\right){g}^{\left({n}\right)} \left({x}\right){dx} \\ $$

Question Number 30559 Answers: 0 Comments: 0

find I_n = ∫_0 ^1 (1−t^2 )^n dt .

$${find}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt}\:\:. \\ $$

Question Number 30557 Answers: 0 Comments: 0

if ϕ convexe and f continue on [a,b] prove that ϕ( (1/(b−a)) ∫_a ^b f(t)dt)≤ (1/(b−a)) ∫_a ^b ϕof(t)dt.

$${if}\:\varphi\:{convexe}\:{and}\:{f}\:{continue}\:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\varphi\left(\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:{f}\left({t}\right){dt}\right)\leqslant\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:\varphi{of}\left({t}\right){dt}. \\ $$

Question Number 30555 Answers: 0 Comments: 0

find ∫_0 ^1 (dt/(√(1−t^4 ))) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{\sqrt{\mathrm{1}−{t}^{\mathrm{4}} }}\:. \\ $$

Question Number 30554 Answers: 0 Comments: 0

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

$${find}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{xcos}\theta\:+\mathrm{1}}{{x}^{\mathrm{2}} \:+\mathrm{2}{xcos}\theta\:+\mathrm{1}}{dx}\:. \\ $$

Question Number 30548 Answers: 0 Comments: 0

let put for ∣λ∣<1 u_n = ∫_0 ^π ((cos(nx))/(1−2λcosx +λ^2 ))dx find u_n interms of n and λ.

$${let}\:{put}\:\:{for}\:\mid\lambda\mid<\mathrm{1}\:\:\:\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{1}−\mathrm{2}\lambda{cosx}\:+\lambda^{\mathrm{2}} }{dx}\: \\ $$$${find}\:{u}_{{n}} \:{interms}\:{of}\:{n}\:{and}\:\lambda. \\ $$

Question Number 30546 Answers: 0 Comments: 0

find I = ∫_1 ^(+∞) (((−1)^([x]) )/x^2 )dx .

$${find}\:{I}\:=\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{\left(−\mathrm{1}\right)^{\left[{x}\right]} }{{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 30544 Answers: 0 Comments: 0

find I= ∫_0 ^π (t/(2+sint))dt.

$${find}\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{t}}{\mathrm{2}+{sint}}{dt}. \\ $$

Question Number 30542 Answers: 0 Comments: 0

prove that ∫_0 ^x e^(−u^2 ) du= x ∫_0 ^(π/4) (e^(−x^2 tan^2 t) /(cos^2 t))dt .

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{{x}} \:\:{e}^{−{u}^{\mathrm{2}} } {du}=\:{x}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{e}^{−{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {t}} }{{cos}^{\mathrm{2}} {t}}{dt}\:\:. \\ $$$$ \\ $$

Question Number 30527 Answers: 1 Comments: 0

find I_(n,p) = ∫_0 ^∞ x^n e^(−px) with n and p from N^★ .

$${find}\:\:{I}_{{n},{p}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{x}^{{n}} \:{e}^{−{px}} \:\:\:\:\:{with}\:{n}\:{and}\:{p}\:{from}\:{N}^{\bigstar} \:. \\ $$

Question Number 30525 Answers: 0 Comments: 0

let I = ∫_0 ^∞ (e^(−x) /(1+x^2 )) give I at form of series .

$${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{give}\:{I}\:{at}\:{form}\:{of}\:{series}\:. \\ $$

Question Number 30521 Answers: 0 Comments: 2

1) find ∫_0 ^1 ((√(1+x^2 )))^n cos(narctanx)dx 2)find ∫_0 ^1 ((√(1+x^2 )))^3 cos(3 arctanx)dx .

$$\left.\mathrm{1}\right)\:{find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)^{{n}} \:{cos}\left({narctanx}\right){dx} \\ $$$$\left.\mathrm{2}\right){find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\right)^{\mathrm{3}} \:{cos}\left(\mathrm{3}\:{arctanx}\right){dx}\:. \\ $$

Question Number 30518 Answers: 1 Comments: 0

let a>0 find f(a) =∫_0 ^∞ (dx/((x+a)(√(a^2 +x^2 )))) .

$${let}\:{a}>\mathrm{0}\:{find}\:\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}+{a}\right)\sqrt{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 30512 Answers: 0 Comments: 1

find I =∫_0 ^1 (√((1−t)/(1+t))) dt .

$${find}\:\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\sqrt{\frac{\mathrm{1}−{t}}{\mathrm{1}+{t}}}\:{dt}\:. \\ $$

Question Number 30508 Answers: 0 Comments: 1

find I= ∫ e^(arcsinx) dx .

$${find}\:{I}=\:\int\:\:{e}^{{arcsinx}} {dx}\:. \\ $$

Question Number 30507 Answers: 0 Comments: 0

find ∫_(−π) ^π (dx/(2+cosx)) 2) if (1/(2+cosx))= (a_0 /2) +Σ_(n≥1) a_n cos(nx) find a_0 and a_n .

$${find}\:\int_{−\pi} ^{\pi} \:\:\frac{{dx}}{\mathrm{2}+{cosx}} \\ $$$$\left.\mathrm{2}\right)\:{if}\:\:\frac{\mathrm{1}}{\mathrm{2}+{cosx}}=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}\geqslant\mathrm{1}} {a}_{{n}} \:{cos}\left({nx}\right)\:\:{find}\:{a}_{\mathrm{0}} \:{and}\:{a}_{{n}} \:. \\ $$

Question Number 30506 Answers: 0 Comments: 0

find f(x) =∫_0 ^x (t/(1+t^4 ))dt with x>0.

$${find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\:{with}\:{x}>\mathrm{0}. \\ $$

Question Number 30500 Answers: 0 Comments: 0

find ∫_1 ^(+∞) (dt/(t^2 (1+t^2 ))) .

$${find}\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{2}} \left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:. \\ $$

Question Number 30499 Answers: 0 Comments: 0

let put F(x)= ∫_0 ^x (√(tant)) dt with x>0 find F(x).

$${let}\:{put}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:\sqrt{{tant}}\:\:{dt}\:{with}\:{x}>\mathrm{0}\:\:{find}\:{F}\left({x}\right). \\ $$

Question Number 30498 Answers: 1 Comments: 0

find I= ∫_0 ^(√3) arcsin(((2x)/(1+x^2 )))dx .

$${find}\:\:{I}=\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:{arcsin}\left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){dx}\:\:. \\ $$

Question Number 30494 Answers: 1 Comments: 0

find I= ∫_0 ^1 (dx/((1+x)(√(1+x^2 )))) .

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

Question Number 30480 Answers: 0 Comments: 0

let f(x)= Σ_(k=2) ^∞ (((−1)^k )/(x+k)) 1) find D_f 2)let put δ(x)= Σ_(n=1) ^∞ (((−1)^n )/n^x ) (Rieman alternate serie) find f(x) interms of δ(x).

$${let}\:{f}\left({x}\right)=\:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{x}+{k}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right){let}\:{put}\:\delta\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} }\:\:\left({Rieman}\:{alternate}\:{serie}\right) \\ $$$${find}\:{f}\left({x}\right)\:{interms}\:{of}\:\delta\left({x}\right). \\ $$

Question Number 30478 Answers: 0 Comments: 0

let give l_i (x)= ∫_2 ^x (dt/(ln(t))) find a serie equal to l_i (x). x≥2.

$${let}\:{give}\:\:{l}_{{i}} \left({x}\right)=\:\int_{\mathrm{2}} ^{{x}} \:\:\:\frac{{dt}}{{ln}\left({t}\right)}\:{find}\:{a}\:{serie}\:{equal}\:{to}\:{l}_{{i}} \left({x}\right). \\ $$$${x}\geqslant\mathrm{2}. \\ $$

Question Number 30477 Answers: 0 Comments: 0

f function 2(×) derivable prove that L(f^′ )= pL(f) −f(o) and L(f^(′′) )=p^2 L(f)−pf(0)−f^′ (0) 2) let f(t)=tsin(wt) find L(f).

$${f}\:{function}\:\mathrm{2}\left(×\right)\:{derivable}\:{prove}\:{that} \\ $$$${L}\left({f}^{'} \right)=\:{pL}\left({f}\right)\:−{f}\left({o}\right)\:{and}\:{L}\left({f}^{''} \right)={p}^{\mathrm{2}} {L}\left({f}\right)−{pf}\left(\mathrm{0}\right)−{f}^{'} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{f}\left({t}\right)={tsin}\left({wt}\right)\:{find}\:{L}\left({f}\right). \\ $$

Question Number 30476 Answers: 1 Comments: 0

find L(cos^2 x) and L(sin^2 x) L is laplace transform.

$${find}\:{L}\left({cos}^{\mathrm{2}} {x}\right)\:{and}\:{L}\left({sin}^{\mathrm{2}} {x}\right)\:{L}\:{is}\:{laplace}\:{transform}. \\ $$

Question Number 30475 Answers: 0 Comments: 0

let give f_n (x)= ∫_(1/n) ^n ((sin(xt))/t) e^(−t) dt 1)find lim_(n→∞) f_n (x) 2)find another form of f_n (x) by calculating f_n ^′ (x).

$${let}\:{give}\:{f}_{{n}} \left({x}\right)=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\frac{{sin}\left({xt}\right)}{{t}}\:{e}^{−{t}} \:{dt} \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow\infty} {f}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{another}\:{form}\:{of}\:{f}_{{n}} \left({x}\right)\:{by}\:{calculating}\:{f}_{{n}} ^{'} \left({x}\right). \\ $$

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