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IntegrationQuestion and Answers: Page 130

Question Number 115725 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((lnx)/((x^(2 ) +x+2)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{lnx}}{\left({x}^{\mathrm{2}\:} +{x}+\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 115689 Answers: 1 Comments: 0

Question Number 115664 Answers: 2 Comments: 0

If lim_(x→p) ((p^x −x^p )/(x^x −p^p )) = 1 then p = ?

$${If}\:\underset{{x}\rightarrow{p}} {\mathrm{lim}}\:\frac{{p}^{{x}} −{x}^{{p}} }{{x}^{{x}} −{p}^{{p}} }\:=\:\mathrm{1}\:{then}\:{p}\:=\:? \\ $$

Question Number 115661 Answers: 2 Comments: 0

(1) ∫ sin ((√x)) dx (2) ∫ cos ((√x) ) dx (3) ∫ tan ((√x) ) dx

$$\left(\mathrm{1}\right)\:\int\:\mathrm{sin}\:\left(\sqrt{{x}}\right)\:{dx} \\ $$$$\left(\mathrm{2}\right)\:\int\:\mathrm{cos}\:\left(\sqrt{{x}}\:\right)\:{dx}\: \\ $$$$\left(\mathrm{3}\right)\:\int\:\mathrm{tan}\:\left(\sqrt{{x}}\:\right)\:{dx}\: \\ $$

Question Number 115656 Answers: 3 Comments: 1

∫ (√((x−a)/(b−x))) dx = ? where a <x < b

$$\int\:\sqrt{\frac{{x}−{a}}{{b}−{x}}}\:{dx}\:=\:? \\ $$$${where}\:{a}\:<{x}\:<\:{b} \\ $$

Question Number 115594 Answers: 1 Comments: 0

old question, I couldn′t find it: ∫(√(x−(√x)))dx=?

$$\mathrm{old}\:\mathrm{question},\:\mathrm{I}\:\mathrm{couldn}'\mathrm{t}\:\mathrm{find}\:\mathrm{it}: \\ $$$$\int\sqrt{{x}−\sqrt{{x}}}{dx}=? \\ $$

Question Number 115592 Answers: 0 Comments: 1

Question Number 115558 Answers: 2 Comments: 2

... advanced calculus... evaluate :: ∫_0 ^( ∞) ln(1+ax^2 )ln(1+(b/x^2 ))dx m.n.july

$$\:\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}...\: \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\mathrm{1}+{ax}^{\mathrm{2}} \right){ln}\left(\mathrm{1}+\frac{{b}}{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\:\:\:\:\:\:\:{m}.{n}.{july} \\ $$$$ \\ $$

Question Number 115533 Answers: 0 Comments: 0

Question Number 115531 Answers: 0 Comments: 0

Question Number 115532 Answers: 0 Comments: 0

Question Number 115507 Answers: 2 Comments: 0

.... ...matematical analysis... prove that ::: a>0 :: [((i : ∫_(0 ) ^( ∞) ((sin^2 (ax))/x^(3/2) ) dx= (√(πa)))),((ii: ∫_0 ^( ∞) ((sin^3 (ax))/( (√x))) dx = ((−1+3(√(3 )))/4) (√((π/(6a)) )) )) ] ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:....\:\:\:...{matematical}\:{analysis}...\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}>\mathrm{0}\:::\:\:\:\begin{bmatrix}{{i}\::\:\:\int_{\mathrm{0}\:} ^{\:\infty} \frac{{sin}^{\mathrm{2}} \left({ax}\right)}{{x}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:{dx}=\:\sqrt{\pi{a}}}\\{{ii}:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({ax}\right)}{\:\sqrt{{x}}}\:{dx}\:=\:\frac{−\mathrm{1}+\mathrm{3}\sqrt{\mathrm{3}\:}}{\mathrm{4}}\:\sqrt{\frac{\pi}{\mathrm{6}{a}}\:\:}\:}\end{bmatrix} \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 115459 Answers: 3 Comments: 0

I= ∫_0 ^1 (dx/((1+x^3 )((1+x^3 ))^(1/(3 )) )) ? I=∫_0 ^(π/2) cos^2 x cos^2 (2x) dx = ?

$${I}=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)\sqrt[{\mathrm{3}\:}]{\mathrm{1}+{x}^{\mathrm{3}} }}\:? \\ $$$${I}=\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{cos}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{2}{x}\right)\:{dx}\:=\:? \\ $$$$ \\ $$

Question Number 115449 Answers: 2 Comments: 0

calculate ∫_(−∞) ^∞ (x^2 /((x^2 −x+1)^3 ))dx

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

Question Number 115464 Answers: 5 Comments: 0

I= ∫_(0 ) ^1 x ln (1+x^2 ) dx ? I=∫ (√(sin x)) .cos^3 x dx ?

$${I}=\:\underset{\mathrm{0}\:} {\overset{\mathrm{1}} {\int}}\:{x}\:\mathrm{ln}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:{dx}\:? \\ $$$${I}=\int\:\sqrt{\mathrm{sin}\:{x}}\:.\mathrm{cos}\:^{\mathrm{3}} {x}\:{dx}\:? \\ $$

Question Number 115367 Answers: 1 Comments: 0

solve xy^(′′) −(x^2 +1)y^′ =x^2 sin(2x)

$${solve}\:{xy}^{''} −\left({x}^{\mathrm{2}} +\mathrm{1}\right){y}^{'} \:\:={x}^{\mathrm{2}} {sin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 115366 Answers: 2 Comments: 0

calculate ∫_(−1) ^2 (dx/(ch^2 x +sh^2 x))

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{2}} \:\frac{{dx}}{{ch}^{\mathrm{2}} {x}\:+{sh}^{\mathrm{2}} {x}} \\ $$

Question Number 115365 Answers: 0 Comments: 0

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

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

Question Number 115364 Answers: 1 Comments: 0

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

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

Question Number 115362 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((sinx)/(1+x^2 ))dx

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

Question Number 115361 Answers: 1 Comments: 0

find ∫_0 ^∞ ((cos(πx^2 ))/((x^2 +3)^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\pi{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 115267 Answers: 1 Comments: 0

If f(x) is a differentiable function defined ∀x∈R such that (f(x))^3 −x+f(x)=0 then ∫_0 ^(√2) f^(−1) (x) dx =

$${If}\:{f}\left({x}\right)\:{is}\:{a}\:{differentiable}\:{function} \\ $$$${defined}\:\:\forall{x}\in\mathbb{R}\:{such}\:{that}\:\left({f}\left({x}\right)\right)^{\mathrm{3}} −{x}+{f}\left({x}\right)=\mathrm{0} \\ $$$${then}\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{2}}} {\int}}\:{f}^{−\mathrm{1}} \left({x}\right)\:{dx}\:=\: \\ $$

Question Number 115193 Answers: 1 Comments: 1

...advanced mathematics... :: digamma limit :: if k>0 then prove that lim_(x→0) (1/x)(ψ(((k+x)/(2x))) − ψ((k/(2x)))) =(1/k) ✓ m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:{mathematics}...\:\: \\ $$$$\:\:\:\:\:\:\:::\:\:\:{digamma}\:\:{limit}\:\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:{if}\:\:\:{k}>\mathrm{0}\:\:{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{1}}{{x}}\left(\psi\left(\frac{{k}+{x}}{\mathrm{2}{x}}\right)\:−\:\psi\left(\frac{{k}}{\mathrm{2}{x}}\right)\right)\:=\frac{\mathrm{1}}{{k}}\:\:\:\:\checkmark \\ $$$$ \\ $$$$\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970}... \\ $$$$\: \\ $$

Question Number 115169 Answers: 3 Comments: 0

∫ sin^2 (ln x) dx

$$\int\:\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{ln}\:{x}\right)\:{dx}\: \\ $$

Question Number 115111 Answers: 2 Comments: 0

...mathematical analysis... prove that: Ω=∫_0 ^( 1) (((x^8 +1)ln(x))/(x^(10) −1)) dx=((π^2 ϕ^2 )/(25)) ✓ m.n.july 1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{mathematical}\:\:{analysis}...\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({x}^{\mathrm{8}} +\mathrm{1}\right){ln}\left({x}\right)}{{x}^{\mathrm{10}} −\mathrm{1}}\:{dx}=\frac{\pi^{\mathrm{2}} \varphi^{\mathrm{2}} }{\mathrm{25}}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}\:\mathrm{1970} \\ $$$$ \\ $$

Question Number 115071 Answers: 1 Comments: 0

∫_0 ^(π/2) ((cos x)/( (√(1−sin x)))) dx ?

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:{x}}}\:{dx}\:? \\ $$

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