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let give the function f(x)=∫_0 ^π ln(1+xcosθ)dθ with ∣x∣<1 1) find a simple form of f(x) 2)calculate ∫_0 ^π ln(1−cosθ)dθ 3)calculate ∫_0 ^π ln(1+cosθ)dθ.

$${let}\:{give}\:{the}\:{function}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}+{xcos}\theta\right){d}\theta\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\mathrm{1}−{cos}\theta\right){d}\theta \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}+{cos}\theta\right){d}\theta. \\ $$

Question Number 31462 Answers: 0 Comments: 0

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

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

Question Number 31460 Answers: 0 Comments: 1

find in terms of n the value of A_n = ∫_0 ^1 Π_(k=1) ^(n−1) (x^2 −2xcos(((kπ)/n)) +1)dx with n from N^★ .

$${find}\:{in}\:{terms}\:{of}\:\:{n}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \left({x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\left(\frac{{k}\pi}{{n}}\right)\:+\mathrm{1}\right){dx}\:\:\:{with}\:{n}\:{from}\:{N}^{\bigstar} . \\ $$

Question Number 31459 Answers: 0 Comments: 0

find ∫_0 ^(π/4) ((cost)/(cos^3 t +sin^3 t)) dt.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{cost}}{{cos}^{\mathrm{3}} {t}\:+{sin}^{\mathrm{3}} {t}}\:{dt}. \\ $$

Question Number 31458 Answers: 0 Comments: 0

calculate ∫_0 ^(√3) arcsin(((2t)/(1+t^2 )))dt .

$${calculate}\:\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:\:{arcsin}\left(\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right){dt}\:. \\ $$

Question Number 31419 Answers: 0 Comments: 0

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

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

Question Number 31418 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctanx)/(x^2 +x+1)) dx.

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctanx}}{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}\:{dx}. \\ $$

Question Number 31415 Answers: 0 Comments: 0

let 0<x<1 find f(x)=∫_0 ^x lnt .ln(1−t)dt.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {lnt}\:.{ln}\left(\mathrm{1}−{t}\right){dt}. \\ $$

Question Number 31414 Answers: 0 Comments: 0

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

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

Question Number 31296 Answers: 0 Comments: 11

find ∫_0 ^(+∞) (dx/((1+x^2 )^n )) with n integr and n≥1 .

$${find}\:\:\int_{\mathrm{0}} ^{+\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1}\:. \\ $$

Question Number 31107 Answers: 0 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^n )) with n>1.

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

Question Number 31106 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−x^2 ) =lim_(n→+∞) ∫_0 ^∞ (dx/((1+x^2 )^n )) . 2) prove that (1/(√π)) =lim_(n→∞) ((1.3.5....(2n−3))/(2.4.6....(2n−2))) (√n) (wallis formula).

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } ={lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\frac{\mathrm{1}}{\sqrt{\pi}}\:={lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}....\left(\mathrm{2}{n}−\mathrm{3}\right)}{\mathrm{2}.\mathrm{4}.\mathrm{6}....\left(\mathrm{2}{n}−\mathrm{2}\right)}\:\sqrt{{n}} \\ $$$$\left({wallis}\:{formula}\right). \\ $$

Question Number 31105 Answers: 0 Comments: 1

prove that ∫_0 ^x e^(−t^2 ) dt =((√π)/2) −(e^(−x^2 ) /(√π)) ∫_0 ^∞ (e^(−x^2 t^2 ) /(1+t^2 )) dt with x>0

$${prove}\:{that}\:\int_{\mathrm{0}} ^{{x}} \:\:\:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:−\frac{{e}^{−{x}^{\mathrm{2}} } }{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} {t}^{\mathrm{2}} } }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 31104 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) e^(−(x^2 +2x−1)) dx .

$${find}\:\:\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{1}\right)} {dx}\:. \\ $$

Question Number 31103 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x −(a/x))^2 ) dx with a≥0 .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({x}\:−\frac{{a}}{{x}}\right)^{\mathrm{2}} } {dx}\:\:{with}\:\:{a}\geqslant\mathrm{0}\:. \\ $$

Question Number 31102 Answers: 0 Comments: 2

find ∫_0 ^(+∞) ((lnx)/(x^2 +a^2 ))dx 2) find the value of ∫_0 ^∞ ((lnx)/((x^2 +a^2 )^3 )) .