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IntegrationQuestion and Answers: Page 132

Question Number 122290 Answers: 1 Comments: 1

... nice calculus... calculate :: Ω=∫_0 ^( 1) x^2 (ψ(1+x)−ψ(2−x))dx=??? .m.n.1970.

$$\:\:\:\:\:...\:\:{nice}\:\:{calculus}... \\ $$$$\:\:\:{calculate}\::: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}} \left(\psi\left(\mathrm{1}+{x}\right)−\psi\left(\mathrm{2}−{x}\right)\right){dx}=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}.\mathrm{1970}. \\ $$

Question Number 122274 Answers: 2 Comments: 0

∫ ((√(1−(√x)))/( (√(1+(√x))))) dx ?

$$\:\:\:\int\:\frac{\sqrt{\mathrm{1}−\sqrt{{x}}}}{\:\sqrt{\mathrm{1}+\sqrt{{x}}}}\:{dx}\:?\: \\ $$

Question Number 122205 Answers: 1 Comments: 2

find ∫ (dx/(x(x+1)(√(x^2 +x))))

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

Question Number 122186 Answers: 2 Comments: 0

Obtain a reduction formulae for I_n = ∫_0 ^1 (ln x)^n dx find I_2 = ∫_0 ^1 (ln x)^2 dx

$$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{reduction}\:\mathrm{formulae}\:\mathrm{for}\: \\ $$$$\:\:\:{I}_{{n}} \:=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\mathrm{ln}\:{x}\right)^{{n}} {dx}\: \\ $$$$\mathrm{find}\:{I}_{\mathrm{2}} =\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}} \:{dx} \\ $$

Question Number 122160 Answers: 1 Comments: 0

Question Number 122159 Answers: 3 Comments: 0

... nice calculus... prove that:: Ω=∫_0 ^( (π/2)) {tan^(−1) (ptan(x))−tan^(−1) (qtan(x))}(tan(x)+cot(x))dx =(π/2) log((p/q)) ( p , q >0 ) m.n.

$$\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:{prove}\:\:{that}:: \\ $$$$\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \left\{{tan}^{−\mathrm{1}} \left({ptan}\left({x}\right)\right)−{tan}^{−\mathrm{1}} \left({qtan}\left({x}\right)\right)\right\}\left({tan}\left({x}\right)+{cot}\left({x}\right)\right){dx} \\ $$$$=\frac{\pi}{\mathrm{2}}\:{log}\left(\frac{{p}}{{q}}\right)\:\:\:\left(\:\:\:\:{p}\:,\:{q}\:>\mathrm{0}\:\:\:\right) \\ $$$$\:\:\:\:{m}.{n}. \\ $$

Question Number 122157 Answers: 2 Comments: 1

find ∫_(−1) ^1 (√(1+x^4 ))dx

$$\mathrm{find}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 122137 Answers: 1 Comments: 0

Question Number 122120 Answers: 2 Comments: 0

∫_0 ^3 (1/( (√y))).e^y dy

$$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{1}}{\:\sqrt{{y}}}.{e}^{{y}} {dy} \\ $$

Question Number 122109 Answers: 2 Comments: 0

Question Number 122108 Answers: 1 Comments: 0

∫_1 ^2 ((ln (x))/x^2 ) dx ?

$$\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{ln}\:\left({x}\right)}{{x}^{\mathrm{2}} }\:{dx}\:? \\ $$

Question Number 122098 Answers: 1 Comments: 1

Question Number 122044 Answers: 0 Comments: 3

Question Number 122035 Answers: 1 Comments: 0

...nice calculus... prove that : ∫_0 ^( 1) ((ln^2 (1+x))/x)dx=^(??) ((ζ(3))/4) .m.n.

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:{calculus}... \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}+{x}\right)}{{x}}{dx}\overset{??} {=}\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}. \\ $$

Question Number 121972 Answers: 1 Comments: 0

∫_0 ^( ∞) e^(−x^2 ) cos(5x)dx

$$\int_{\mathrm{0}} ^{\:\infty} \mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{cos}\left(\mathrm{5x}\right)\mathrm{dx} \\ $$

Question Number 121965 Answers: 2 Comments: 0

∫_4 ^9 ((ℓn (x))/( (√x))) dx ?

$$\:\:\:\:\:\underset{\mathrm{4}} {\overset{\mathrm{9}} {\int}}\:\frac{\ell{n}\:\left({x}\right)}{\:\sqrt{{x}}}\:{dx}\:? \\ $$

Question Number 121962 Answers: 2 Comments: 0

∫_0 ^1 (dx/( ((x^2 −x^3 ))^(1/(3 )) )) ?

$$\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{dx}}{\:\sqrt[{\mathrm{3}\:}]{{x}^{\mathrm{2}} −{x}^{\mathrm{3}} }}\:?\: \\ $$

Question Number 121928 Answers: 1 Comments: 2

Σ_(n=0) ^∞ ((1/(12n+1))+(1/(12n+5))−(1/(12n+7))−(1/(12n+11))) Problem source : Brilliant.Org

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{12}{n}+\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{12}{n}+\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{12}{n}+\mathrm{7}}−\frac{\mathrm{1}}{\mathrm{12}{n}+\mathrm{11}}\right) \\ $$$$ \\ $$$${Problem}\:{source}\::\:{Brilliant}.{Org} \\ $$

Question Number 121887 Answers: 2 Comments: 2

Question Number 121886 Answers: 2 Comments: 1

Question Number 121875 Answers: 2 Comments: 0

Question Number 121862 Answers: 1 Comments: 0

1)explicite f(a)=∫_0 ^∞ ((t^(a−1) lnt)/(1+t))dt with 0<a<1 2)calculate ∫_0 ^∞ ((lnt)/((1+t)(√t)))dt

$$\left.\mathrm{1}\right){explicite}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{t}^{{a}−\mathrm{1}} {lnt}}{\mathrm{1}+{t}}{dt} \\ $$$${with}\:\mathrm{0}<{a}<\mathrm{1} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left(\mathrm{1}+{t}\right)\sqrt{{t}}}{dt} \\ $$

Question Number 121860 Answers: 0 Comments: 0

calculate ∫∫_([0,1]^2 ) ((arctan((√(x^2 +y^2 ))))/(x^2 +y^2 ))dxdy

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\frac{{arctan}\left(\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{dxdy} \\ $$

Question Number 121859 Answers: 2 Comments: 0

find ∫ (dx/((x−1)(√(x+1))−(x+1)(√(x−1))))

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

Question Number 121858 Answers: 0 Comments: 0

decompose F(x) =(x^n /(x^(2n+1) +1)) 2)find the value of ∫_0 ^∞ (x^n /(x^(2n+1) +1))dx n integr natural

$${decompose}\:{F}\left({x}\right)\:=\frac{{x}^{{n}} }{{x}^{\mathrm{2}{n}+\mathrm{1}} +\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{{n}} }{{x}^{\mathrm{2}{n}+\mathrm{1}} +\mathrm{1}}{dx} \\ $$$${n}\:{integr}\:{natural} \\ $$

Question Number 121857 Answers: 0 Comments: 0

find ∫_0 ^∞ xe^(−x^2 ) arctan(2x)dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{xe}^{−{x}^{\mathrm{2}} } {arctan}\left(\mathrm{2}{x}\right){dx} \\ $$

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