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Question Number 37353 Answers: 0 Comments: 1

let g(z) =(z/(e^z −1)) developp g at integr serie .

$${let}\:{g}\left({z}\right)\:=\frac{{z}}{{e}^{{z}} −\mathrm{1}} \\ $$$${developp}\:{g}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 37352 Answers: 0 Comments: 1

let f(z) = e^(−(1/z^2 )) 1) give f(z) at form of serie 2) give ∫_1 ^2 f(z)dz at form of serie .

$${let}\:\:{f}\left({z}\right)\:=\:{e}^{−\frac{\mathrm{1}}{{z}^{\mathrm{2}} }} \:\: \\ $$$$\left.\mathrm{1}\right)\:{give}\:{f}\left({z}\right)\:{at}\:{form}\:{of}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{give}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {f}\left({z}\right){dz}\:\:\:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 37350 Answers: 0 Comments: 2

fond the value of ∫_0 ^(2π) (dt/((a +cost)^2 )) with a>1.

$${fond}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{\left({a}\:+{cost}\right)^{\mathrm{2}} }\:\:{with}\:{a}>\mathrm{1}. \\ $$$$ \\ $$

Question Number 37349 Answers: 0 Comments: 1

calculate ∫_0 ^(2π) (dt/(1−2pcost +p^2 )) if ∣p∣<1

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{\mathrm{1}−\mathrm{2}{pcost}\:+{p}^{\mathrm{2}} }\:\:{if}\:\mid{p}\mid<\mathrm{1} \\ $$

Question Number 37348 Answers: 0 Comments: 2

calculate ∫_0 ^(2π) (dt/(p +cost)) with p>1

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{dt}}{{p}\:+{cost}}\:\:{with}\:{p}>\mathrm{1} \\ $$

Question Number 37347 Answers: 1 Comments: 2

let r =(√(p^2 +q^2 )) p and q from R and p>0 q>0 1)prove that ∫_0 ^(+∞) e^(−px) ((cos(px))/(√x))dx=((√π)/r)(√((r+p)/2)) 2) ∫_0 ^∞ e^(−px) ((sin(qx))/(√x))dx =((√π)/r) (√((r−p)/2))

$${let}\:{r}\:=\sqrt{{p}^{\mathrm{2}} \:+{q}^{\mathrm{2}} }\:\:\:{p}\:{and}\:{q}\:{from}\:{R}\:\:{and}\:{p}>\mathrm{0}\:\:{q}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:{e}^{−{px}} \:\frac{{cos}\left({px}\right)}{\sqrt{{x}}}{dx}=\frac{\sqrt{\pi}}{{r}}\sqrt{\frac{{r}+{p}}{\mathrm{2}}} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{px}} \:\:\frac{{sin}\left({qx}\right)}{\sqrt{{x}}}{dx}\:=\frac{\sqrt{\pi}}{{r}}\:\sqrt{\frac{{r}−{p}}{\mathrm{2}}} \\ $$

Question Number 37346 Answers: 0 Comments: 0

find the value of ∫_0 ^(2π) (dt/(a cos^2 t +b sin^2 t)) with a>0 and b>0 .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{a}\:{cos}^{\mathrm{2}} {t}\:+{b}\:{sin}^{\mathrm{2}} {t}} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:. \\ $$

Question Number 37345 Answers: 0 Comments: 0

calculate I(a) = ∫_0 ^(2π) ((1+acost)/(1+2acost +a^2 ))dt 1) if ∣a∣<1 2) if ∣a∣>1

$${calculate}\:{I}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{\mathrm{1}+{acost}}{\mathrm{1}+\mathrm{2}{acost}\:+{a}^{\mathrm{2}} }{dt}\:\: \\ $$$$\left.\mathrm{1}\right)\:{if}\:\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{if}\:\mid{a}\mid>\mathrm{1} \\ $$

Question Number 37344 Answers: 0 Comments: 1

solve the d.e. y^′ −xy =cosx .

$${solve}\:{the}\:{d}.{e}.\:{y}^{'} \:−{xy}\:\:={cosx}\:. \\ $$

Question Number 37343 Answers: 0 Comments: 3

let f(x) = ∫_0 ^1 ln(1+xt^2 )dt with ∣x∣<1 1) find f(x) 2) calculate ∫_0 ^1 ln(2+t^2 )dt .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right){dt}\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{2}+{t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 37342 Answers: 0 Comments: 2

calculate Σ_(n=1) ^∞ (((−1)^n )/(n^2 (n+1))) x^n with ∣x∣<1 2) find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)2^n )) .

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:{x}^{{n}} \:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 37341 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (3/(n^2 (2n+1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{3}}{{n}^{\mathrm{2}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 37339 Answers: 0 Comments: 1

find the value of Σ_(n=1) ^∞ ((2n+1)/(1 +2^3 +3^3 +...+n^3 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\:\:\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{1}\:+\mathrm{2}^{\mathrm{3}} \:+\mathrm{3}^{\mathrm{3}} \:+...+{n}^{\mathrm{3}} } \\ $$

Question Number 37338 Answers: 0 Comments: 1

calculate B_n = ∫_0 ^1 sh^n xdx .

$${calculate}\:\:{B}_{{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{sh}^{{n}} {xdx}\:. \\ $$

Question Number 37337 Answers: 0 Comments: 1

calculate A_n = ∫_0 ^1 ch^n xdx .

$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ch}^{{n}} {xdx}\:. \\ $$

Question Number 37335 Answers: 0 Comments: 1

find ∫ x arctan(x+(1/x))dx .

$${find}\:\int\:\:\:\:\:{x}\:{arctan}\left({x}+\frac{\mathrm{1}}{{x}}\right){dx}\:. \\ $$

Question Number 37334 Answers: 0 Comments: 1

study the convergence of Σ_(n=0) ^∞ e^(−x) Σ_(k=0) ^∞ (x^k /(k!)) .

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \:\sum_{{k}=\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{k}} }{{k}!}\:. \\ $$

Question Number 37333 Answers: 0 Comments: 2

let f(x)=Σ_(n=1) ^∞ ((sin(nx))/n^3 ) 1)study the convergence of this serie 2)prove that ∫_0 ^π f(x)dx=2 Σ_(n=1) ^∞ (1/((2n−1)^4 )) 3)prove that ∀x∈ ∈R f^′ (x)=Σ_(n=1) ^∞ ((cos(nx))/n^2 ) 4) prove that ∫_0 ^(π/2) ( Σ_(n≥1) ((cos(nx))/n^2 ))=Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^2 ))

$${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{3}} } \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}=\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:\forall{x}\in\:\in{R}\:\:{f}^{'} \left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\:\sum_{{n}\geqslant\mathrm{1}} \frac{{cos}\left({nx}\right)}{{n}^{\mathrm{2}} }\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 37324 Answers: 0 Comments: 3