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IntegrationQuestion and Answers: Page 147

Question Number 112613 Answers: 1 Comments: 2

....number theory... Question : If a , b , c ∈ N ; then prove ::: a!∗b!∗c!∣(a+b+c)! m.n . july 970#

$$\:\:\:\:\:....{number}\:{theory}... \\ $$$$\:\:\:\:\:\:\:{Question}\::\:\:\:\:\:\:\mathrm{I}{f}\:\:\:{a}\:,\:{b}\:,\:{c}\:\:\in\:\mathbb{N}\:\:\:;\:{then}\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}!\ast{b}!\ast{c}!\mid\left({a}+{b}+{c}\right)!\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}\:.\:{july}\:\mathrm{970}# \\ $$

Question Number 112579 Answers: 4 Comments: 0

.... calculus .... Please Evaluate ::: Ω = ∫_0 ^(π/2) ((x/(sin(x))))^2 dx =??? M.N.july 1970#

$$\:\:\:\:\:\:\:\:....\:{calculus}\:.... \\ $$$$\mathscr{P}{lease}\:\:\mathscr{E}{valuate}\:::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\frac{{x}}{{sin}\left({x}\right)}\right)^{\mathrm{2}} {dx}\:=???\:\:\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{M}.\mathscr{N}.{july}\:\mathrm{1970}# \\ $$$$\: \\ $$

Question Number 112520 Answers: 0 Comments: 0

∫(((cos(x)))^(1/3) /(cos(x)))dx

$$\int\frac{\sqrt[{\mathrm{3}}]{{cos}\left({x}\right)}}{{cos}\left({x}\right)}{dx} \\ $$

Question Number 112498 Answers: 0 Comments: 0

∫((ln(1+x)dx)/((1+x^2 )))

$$\int\frac{{ln}\left(\mathrm{1}+{x}\right){dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)} \\ $$

Question Number 112481 Answers: 1 Comments: 0

∫ ((sin 2x)/( (√(1+cos^2 x)))) dx

$$\:\int\:\frac{\mathrm{sin}\:\mathrm{2x}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}}}\:\mathrm{dx}\: \\ $$

Question Number 112447 Answers: 0 Comments: 0

calculate ∫_0 ^1 ((ln(1+x^2 ))/(1+x^3 )) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx} \\ $$

Question Number 112446 Answers: 2 Comments: 0

find ∫_0 ^1 ((ln(1+x^2 ))/(1+x))dx

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

Question Number 112449 Answers: 3 Comments: 3

1)calculste A=∫_(−∞) ^(+∞) (dx/((x^2 −ix +1)^2 )) 2) extract Re(A) and Im(A) and determines its values

$$\left.\mathrm{1}\right)\mathrm{calculste}\:\:\mathrm{A}=\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{ix}\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{extract}\:\mathrm{Re}\left(\mathrm{A}\right)\:\mathrm{and}\:\mathrm{Im}\left(\mathrm{A}\right)\:\mathrm{and}\:\mathrm{determines}\:\mathrm{its}\:\mathrm{values} \\ $$

Question Number 112381 Answers: 0 Comments: 0

∫sin(x^3 ) dx

$$\int{sin}\left({x}^{\mathrm{3}} \right)\:{dx} \\ $$

Question Number 112271 Answers: 2 Comments: 0

(√(bemath)) ∫ sin x (√(1−sin x)) dx ?

$$\:\:\:\:\sqrt{\mathrm{bemath}} \\ $$$$\:\:\:\int\:\mathrm{sin}\:\mathrm{x}\:\sqrt{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$

Question Number 112251 Answers: 1 Comments: 0

∫ (dx/((x^4 −1)(√(x^2 +1))))

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

Question Number 112249 Answers: 2 Comments: 1

∫(dx/((αx^2 +px+β)(√(αx^2 +qx+β))))=?

$$\int\frac{{dx}}{\left(\alpha{x}^{\mathrm{2}} +{px}+\beta\right)\sqrt{\alpha{x}^{\mathrm{2}} +{qx}+\beta}}=? \\ $$

Question Number 112206 Answers: 0 Comments: 0

∫ln(sin(x)) dx

$$\int{ln}\left({sin}\left({x}\right)\right)\:{dx} \\ $$

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

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

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