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IntegrationQuestion and Answers: Page 149

Question Number 112189 Answers: 0 Comments: 0

prove that ∫_0 ^∞ (1/(cos(x)+sinh(x)))dx=1.4917.

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{{cos}\left({x}\right)+{sinh}\left({x}\right)}{dx}=\mathrm{1}.\mathrm{4917}. \\ $$

Question Number 112169 Answers: 2 Comments: 0

∫ tan^3 x sec^3 x dx ?

$$\:\int\:\mathrm{tan}\:^{\mathrm{3}} {x}\:\mathrm{sec}\:^{\mathrm{3}} {x}\:{dx}\:? \\ $$

Question Number 112119 Answers: 1 Comments: 0

....calculus.... prove that::: if Ω =∫_(0 ) ^( 1) ln(ln(1−(√x) ))dx then Re(Ω) := −γ + ln(2).... m.n. july 1970#

$$\:\:\:\:\:\:\:\:\:\:\:\:\:....{calculus}.... \\ $$$${prove}\:{that}::: \\ $$$${if}\:\:\:\Omega\:=\int_{\mathrm{0}\:\:} ^{\:\mathrm{1}} {ln}\left({ln}\left(\mathrm{1}−\sqrt{{x}}\:\right)\right){dx} \\ $$$${then} \\ $$$$\mathscr{R}{e}\left(\Omega\right)\::=\:−\gamma\:+\:{ln}\left(\mathrm{2}\right).... \\ $$$$ \\ $$$${m}.{n}.\:{july}\:\mathrm{1970}# \\ $$

Question Number 112545 Answers: 2 Comments: 2

please solve : I=∫_0 ^( 1) xlog^2 (((1−x)/(1+x)))dx =??? ...m.n.july 1970.... good luck .

$$\:\:\:\:{please}\:{solve}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\mathrm{1}} {xlog}^{\mathrm{2}} \left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right){dx}\:=??? \\ $$$$ \\ $$$$\:\:\:\:\:\:...{m}.{n}.{july}\:\mathrm{1970}.... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{good}\:\:\:{luck}\:. \\ $$$$ \\ $$

Question Number 111876 Answers: 0 Comments: 0

Question Number 111873 Answers: 1 Comments: 0

(√(bemath)) ∫ ((2−cos x)/(2+cos x)) dx

$$\:\:\:\sqrt{{bemath}} \\ $$$$\int\:\frac{\mathrm{2}−\mathrm{cos}\:{x}}{\mathrm{2}+\mathrm{cos}\:{x}}\:{dx}\: \\ $$

Question Number 111859 Answers: 2 Comments: 7

I=∫(dx/((x^2 +2x+3)(√(x^2 +x+3)))) = ? my try..

$${I}=\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}\right)\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{3}}}\:=\:? \\ $$$${my}\:{try}.. \\ $$

Question Number 111818 Answers: 3 Comments: 0

(√(bemath )) (1)∫ ((cos x)/(2−cos x)) dx (2) f(x) = ∣x^3 ∣ ⇒ f ′(x) ?

$$\:\:\:\sqrt{{bemath}\:} \\ $$$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{cos}\:{x}}{\mathrm{2}−\mathrm{cos}\:{x}}\:{dx}\: \\ $$$$\left(\mathrm{2}\right)\:{f}\left({x}\right)\:=\:\mid{x}^{\mathrm{3}} \mid\:\Rightarrow\:{f}\:'\left({x}\right)\:? \\ $$

Question Number 111771 Answers: 0 Comments: 0

calculate lim_(n→+∞) a_n ∫_0 ^1 x^(2n) sin(((πx)/2))dx with a_n =Σ_(k=1) ^n sin(((πk)/(2n)))

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{a}_{\mathrm{n}} \int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{2n}} \mathrm{sin}\left(\frac{\pi\mathrm{x}}{\mathrm{2}}\right)\mathrm{dx}\:\mathrm{with}\:\mathrm{a}_{\mathrm{n}} =\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{sin}\left(\frac{\pi\mathrm{k}}{\mathrm{2n}}\right) \\ $$

Question Number 111770 Answers: 0 Comments: 0

f function continue on [0,1] find lim_(n→+∞) (1/n)Σ_(k=0) ^n (n−k)∫_(k/n) ^((k+1)/n) f(x)dx

$$\mathrm{f}\:\mathrm{function}\:\mathrm{continue}\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{\mathrm{n}}\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{n}−\mathrm{k}\right)\int_{\frac{\mathrm{k}}{\mathrm{n}}} ^{\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 111768 Answers: 0 Comments: 0

explicite f(x) =∫_0 ^(2π) ln(x^2 −2xcosθ +1)dθ (x≠+^− 1)

$$\mathrm{explicite}\:\mathrm{f}\left(\mathrm{x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)\mathrm{d}\theta\:\:\:\:\:\:\:\:\left(\mathrm{x}\neq\overset{−} {+}\mathrm{1}\right) \\ $$

Question Number 111762 Answers: 1 Comments: 0

caoculate ∫_0 ^(π/4) ln(1+2tanx)dx

$$\mathrm{caoculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{1}+\mathrm{2tanx}\right)\mathrm{dx} \\ $$

Question Number 111760 Answers: 0 Comments: 0

find lim_(x→1^+ ) ∫_x ^x^2 ((ln(t))/((t−1)^2 ))dx

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{1}^{+} } \:\:\:\int_{\mathrm{x}} ^{\mathrm{x}^{\mathrm{2}} } \:\:\frac{\mathrm{ln}\left(\mathrm{t}\right)}{\left(\mathrm{t}−\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 111756 Answers: 0 Comments: 0

find I_n =∫_0 ^(π/4) (du/(cos^n u))

$$\mathrm{find}\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{du}}{\mathrm{cos}^{\mathrm{n}} \mathrm{u}} \\ $$

Question Number 111719 Answers: 2 Comments: 0

....advanced mathematics.... please demonstrate that:: Φ =∫_0 ^( 1) xlog(1−x).log(1+x)= (1/4) − log(2) ... m.n.july 1970 #

$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:....{advanced}\:\:{mathematics}....\: \\ $$$$ \\ $$$${please}\:\:{demonstrate}\:{that}:: \\ $$$$\: \\ $$$$\Phi\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} {xlog}\left(\mathrm{1}−{x}\right).{log}\left(\mathrm{1}+{x}\right)=\:\frac{\mathrm{1}}{\mathrm{4}}\:−\:{log}\left(\mathrm{2}\right)\:\:... \\ $$$$ \\ $$$$\:\:\:\:\:\:{m}.{n}.{july}\:\mathrm{1970}\:# \\ $$$$ \\ $$

Question Number 111558 Answers: 1 Comments: 9

∫(x)^(1/x) dx=?

$$\int\sqrt[{{x}}]{{x}}{dx}=? \\ $$

Question Number 111499 Answers: 0 Comments: 0

Question Number 111429 Answers: 1 Comments: 0

please evaluate : .... I=∫_0 ^( (π/2)) ((1/(ln(tan(x)))) + (1/(1−tan(x))))dx =??? ::: M. N.july 1970 :::

$$\:\:\:\:\:\:{please}\:\:{evaluate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:....\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left(\frac{\mathrm{1}}{{ln}\left({tan}\left({x}\right)\right)}\:+\:\frac{\mathrm{1}}{\mathrm{1}−{tan}\left({x}\right)}\right){dx}\:=??? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\::::\:\:\:\:\mathscr{M}.\:\mathscr{N}.{july}\:\mathrm{1970}\:::: \\ $$$$\:\: \\ $$

Question Number 111357 Answers: 0 Comments: 0

∫_0 ^1 ((tan^(−1) x)/(1+x^3 ))dx

$$\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{tan}^{−\mathrm{1}} {x}}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 111195 Answers: 2 Comments: 0

(√(bemath)) ∫ (dx/( ((x−1))^(1/(3 )) (((x+1)^2 ))^(1/(3 )) )) ?

$$\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\:\:\:\int\:\frac{\mathrm{dx}}{\:\sqrt[{\mathrm{3}\:}]{\mathrm{x}−\mathrm{1}}\:\:\sqrt[{\mathrm{3}\:}]{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }}\:? \\ $$

Question Number 111189 Answers: 4 Comments: 0

(1) ∫ (((x+1)dx)/(x^4 (x−1))) ? (2) (dy/dx) + (y/(x−2)) = 5(x−2)(√y)

$$\:\left(\mathrm{1}\right)\:\:\:\:\:\:\:\int\:\frac{\left({x}+\mathrm{1}\right){dx}}{{x}^{\mathrm{4}} \left({x}−\mathrm{1}\right)}\:?\: \\ $$$$\:\left(\mathrm{2}\right)\:\:\:\:\:\:\frac{{dy}}{{dx}}\:+\:\frac{{y}}{{x}−\mathrm{2}}\:=\:\mathrm{5}\left({x}−\mathrm{2}\right)\sqrt{{y}}\: \\ $$

Question Number 111174 Answers: 0 Comments: 0

following the newest trend − what do I say!? − ahead of it, of course! I post this answer to one of the next questions, look out for it so you won′t miss it! I=I_1 −2I_2 =ξ(5)+Γ(7/3)−2/(√π)+C

$${following}\:{the}\:{newest}\:{trend}\:−\:{what}\:{do}\:{I} \\ $$$${say}!?\:−\:{ahead}\:{of}\:{it},\:{of}\:{course}!\:{I}\:{post}\:{this} \\ $$$${answer}\:{to}\:{one}\:{of}\:{the}\:{next}\:{questions},\:{look} \\ $$$${out}\:{for}\:{it}\:{so}\:{you}\:{won}'{t}\:{miss}\:{it}! \\ $$$$ \\ $$$${I}={I}_{\mathrm{1}} −\mathrm{2}{I}_{\mathrm{2}} =\xi\left(\mathrm{5}\right)+\Gamma\left(\mathrm{7}/\mathrm{3}\right)−\mathrm{2}/\sqrt{\pi}+{C} \\ $$

Question Number 111109 Answers: 0 Comments: 0

Question Number 111104 Answers: 2 Comments: 2

Question Number 111048 Answers: 0 Comments: 0

Question Number 111083 Answers: 2 Comments: 0

(√(bemath)) (1)∫ (dx/(3sin x+sin^3 x)) (2) lim_(x→∞) x(5^(1/x) −1) (3) find the asymptotes (x^2 /a^2 ) − (y^2 /b^2 ) = 1

$$\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{dx}}{\mathrm{3sin}\:\mathrm{x}+\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}} \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{x}\left(\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{x}}} \:−\mathrm{1}\right)\: \\ $$$$\left(\mathrm{3}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{asymptotes}\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} }\:−\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{b}^{\mathrm{2}} }\:=\:\mathrm{1}\: \\ $$

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