Question and Answers Forum

A = Σ_(n=2) ^(2017) [∫_1 ^n 2tan^(−1) x + sin^(−1) (((2x)/(1 + x^2 ))) dx] B = Π_(n=2) ^(2017) [∫_1 ^n 2tan^(−1) x + sin^(−1) (((2x)/(1 + x^2 ))) dx] A + B = ...

$${A}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\sum}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:{dx}\right] \\ $$$${B}\:=\:\underset{{n}=\mathrm{2}} {\overset{\mathrm{2017}} {\prod}}\:\left[\underset{\mathrm{1}} {\overset{{n}} {\int}}\:\mathrm{2tan}^{−\mathrm{1}} \:{x}\:+\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:{dx}\right] \\ $$$${A}\:+\:{B}\:=\:... \\ $$

Question Number 33544 Answers: 0 Comments: 0

1) find the value of ∫_0 ^∞ (( e^(−tx^2 ) )/(1+x^2 )) dx with t >0 2) find the value of ∫_0 ^∞ (((1−e^(−x^2 ) ))/(x^2 (1+x^2 )))dx .

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\:{e}^{−{tx}^{\mathrm{2}} } }{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{with}\:{t}\:>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}−{e}^{−{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\:. \\ $$

Question Number 33531 Answers: 0 Comments: 16

∫_0 ^∞ (e^(−x^2 ) /(x^2 +1))dx=?

$$\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}^{\mathrm{2}} } }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=? \\ $$

Question Number 33507 Answers: 1 Comments: 0

∫((2cos x)/(3−cos 2x))dx=?

$$\int\frac{\mathrm{2cos}\:{x}}{\mathrm{3}−\mathrm{cos}\:\mathrm{2}{x}}{dx}=? \\ $$

Question Number 33362 Answers: 0 Comments: 1

calculate by residus theorem I = ∫_(−∞) ^(+∞) ((cos(πx))/((1+x +x^2 )))dx .

$${calculate}\:{by}\:{residus}\:{theorem} \\ $$$${I}\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\pi{x}\right)}{\left(\mathrm{1}+{x}\:+{x}^{\mathrm{2}} \right)}{dx}\:. \\ $$

Question Number 33357 Answers: 0 Comments: 0

find the rsdius of convergence for the serie Σ_(n=1) ^∞ (1 +(1/n))^n^2 x^n

$${find}\:{the}\:{rsdius}\:{of}\:{convergence}\:{for} \\ $$$${the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}\:+\frac{\mathrm{1}}{{n}}\right)^{{n}^{\mathrm{2}} } \:{x}^{{n}} \: \\ $$$$ \\ $$

Question Number 33356 Answers: 0 Comments: 0

find the radius of Σ_(n≥0) ((n^2 +n)/(2^n +n!)) x^n

$${find}\:{the}\:{radius}\:{of}\:\sum_{{n}\geqslant\mathrm{0}} \frac{{n}^{\mathrm{2}} \:+{n}}{\mathrm{2}^{{n}} \:+{n}!}\:{x}^{{n}} \\ $$

Question Number 33353 Answers: 0 Comments: 1

let x∈]1,+∞[ andλ ∈[−1,1] give the integral ∫_0 ^∞ ((t^(x−1) e^(−t) )/(1−λe^(−t) )) dt at form of serie.

$$\left.{let}\:{x}\in\right]\mathrm{1},+\infty\left[\:{and}\lambda\:\in\left[−\mathrm{1},\mathrm{1}\right]\right. \\ $$$${give}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} }{\mathrm{1}−\lambda{e}^{−{t}} }\:{dt} \\ $$$${at}\:{form}\:{of}\:{serie}. \\ $$

Question Number 33352 Answers: 0 Comments: 1

let give S(x)=Σ_(n≥0) (((−1)^n )/(√(x+n))) ,x>0 1)study the contnuity ,derivsbility,limits at 0^+ and +∞ 2) we give ∫_0 ^∞ e^(−t^2 ) dt =((√π)/2) .prove that ∀ x>0 S(x)=(1/(√π)) ∫_0 ^∞ (e^(−tx) /((√t)(1+e^(−t) )))dt .

$${let}\:{give}\:{S}\left({x}\right)=\sum_{{n}\geqslant\mathrm{0}} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\sqrt{{x}+{n}}}\:,{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{contnuity}\:,{derivsbility},{limits} \\ $$$${at}\:\mathrm{0}^{+} \:{and}\:+\infty \\ $$$$\left.\mathrm{2}\right)\:{we}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:.{prove}\:{that} \\ $$$$\forall\:{x}>\mathrm{0}\:\:{S}\left({x}\right)=\frac{\mathrm{1}}{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}} }{\sqrt{{t}}\left(\mathrm{1}+{e}^{−{t}} \right)}{dt}\:. \\ $$

Question Number 33351 Answers: 0 Comments: 1

prove that ∫_0 ^1 (((−lnx)^p )/(1+x^2 )) =p! Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^(p+1) )) p integr.

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\left(−{lnx}\right)^{{p}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:={p}!\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{{p}+\mathrm{1}} } \\ $$$${p}\:{integr}. \\ $$

Question Number 33350 Answers: 0 Comments: 1

prove that ∫_0 ^∞ ((cos(αx))/(chx))dx= 2 Σ_(n=0) ^∞ (−1)^n ((2n+1)/((2n+1)^2 +α^2 )) .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\alpha{x}\right)}{{chx}}{dx}= \\ $$$$\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:\frac{\mathrm{2}{n}+\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} \:+\alpha^{\mathrm{2}} }\:. \\ $$

Question Number 33349 Answers: 0 Comments: 1

prove that ∫_0 ^∞ x(x−ln(e^x −1))dx=Σ_(n=1) ^∞ (1/n^3 )

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{x}\left({x}−{ln}\left({e}^{{x}} −\mathrm{1}\right)\right){dx}=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} } \\ $$

Question Number 33346 Answers: 0 Comments: 0

let a and b from R /a<b f [a,b]→C continje prove that ∀n ∈N ∫_a ^b (Π_(k=0) ^(n−1) (x+k))f(x)dx=0 ⇒ f=0

$${let}\:{a}\:{and}\:{b}\:{from}\:{R}\:/{a}<{b}\:\:{f}\:\:\left[{a},{b}\right]\rightarrow{C}\:{continje} \\ $$$${prove}\:{that}\:\:\forall{n}\:\in{N}\:\:\int_{{a}} ^{{b}} \left(\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({x}+{k}\right)\right){f}\left({x}\right){dx}=\mathrm{0} \\ $$$$\Rightarrow\:{f}=\mathrm{0} \\ $$

Pg 290 Pg 291 Pg 292 Pg 293 Pg 294 Pg 295 Pg 296 Pg 297 Pg 298 Pg 299

Contact: info@tinkutara.com