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IntegrationQuestion and Answers: Page 125

Question Number 124906    Answers: 0   Comments: 1

∫sinx^3 dx=?

$$\int\boldsymbol{{sinx}}^{\mathrm{3}} \boldsymbol{{dx}}=? \\ $$

Question Number 124903    Answers: 1   Comments: 0

Question Number 124888    Answers: 3   Comments: 0

::::: prove that :::: φ=∫_0 ^( ∞) ((arctan(x^2 ))/x^2 )dx=(π/( (√2)))

$$:::::\:\:{prove}\:{that}\: \\ $$$$\:\:::::\:\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\infty} \frac{{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}=\frac{\pi}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 124887    Answers: 1   Comments: 0

...nice calculus.. evaluate : 2∫_1 ^( ∞) ((({x}−(1/2))/x))dx−∫_0 ^( 1) ln(Γ(x))dx=??? {x}: fractional part...

$$\:\:\:\:\:...{nice}\:\:{calculus}.. \\ $$$$\:\:\:{evaluate}\:: \\ $$$$\:\:\mathrm{2}\int_{\mathrm{1}} ^{\:\infty} \left(\frac{\left\{{x}\right\}−\frac{\mathrm{1}}{\mathrm{2}}}{{x}}\right){dx}−\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){dx}=??? \\ $$$$\left\{{x}\right\}:\:{fractional}\:{part}... \\ $$

Question Number 124827    Answers: 2   Comments: 0

.... nice calculus ... prove that:: ∫_0 ^( (π/2)) ((log(1+tan(x)))/(tan(x)))dx=((5π^2 )/(48)) ✓

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:{nice}\:\:\:{calculus}\:... \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{log}\left(\mathrm{1}+{tan}\left({x}\right)\right)}{{tan}\left({x}\right)}{dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{48}}\:\checkmark \\ $$$$ \\ $$

Question Number 124825    Answers: 0   Comments: 2

Question Number 124785    Answers: 0   Comments: 0

∫_( 0) ^( a) ∫_( 0) ^( (√(a^2 −x^2 ))) (1/((1+e^y )(√(a^2 −x^2 −y^2 ))))dxdy

$$\int_{\:\mathrm{0}} ^{\:\mathrm{a}} \int_{\:\mathrm{0}} ^{\:\sqrt{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }} \frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{e}^{\mathrm{y}} \right)\sqrt{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} }}\mathrm{dxdy} \\ $$$$ \\ $$

Question Number 124794    Answers: 1   Comments: 0

If f(x)= { ((2x ; 0<x<1)),((3 ; x=1 )),((6x−1 ; 1<x<2)) :} find ∫_0 ^2 f(x) dx ?

$${If}\:{f}\left({x}\right)=\begin{cases}{\mathrm{2}{x}\:;\:\mathrm{0}<{x}<\mathrm{1}}\\{\mathrm{3}\:;\:{x}=\mathrm{1}\:}\\{\mathrm{6}{x}−\mathrm{1}\:;\:\mathrm{1}<{x}<\mathrm{2}}\end{cases} \\ $$$${find}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:? \\ $$

Question Number 124853    Answers: 2   Comments: 0

∫_1 ^( x^3 +5x) f(t) dt = 2x then f(18) =?

$$\:\:\int_{\mathrm{1}} ^{\:{x}^{\mathrm{3}} +\mathrm{5}{x}} {f}\left({t}\right)\:{dt}\:=\:\mathrm{2}{x}\:\: \\ $$$$\:{then}\:{f}\left(\mathrm{18}\right)\:=? \\ $$

Question Number 124742    Answers: 0   Comments: 2

∫((3^t +11)/(6^t +11))dt collected problem

$$\int\frac{\mathrm{3}^{{t}} +\mathrm{11}}{\mathrm{6}^{{t}} +\mathrm{11}}{dt}\:\:\:\boldsymbol{{collected}}\:\boldsymbol{{problem}} \\ $$

Question Number 124738    Answers: 0   Comments: 3

...nice calculus... simple limit:: lim_(n→∞) {((1^(a+1) +2^(a+1) +...+n^(a+1) )/(n(1^a +2^a +....n^a )))}=? where a ≠−2 , −1

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:{simple}\:{limit}:: \\ $$$$\:\:\:{lim}_{{n}\rightarrow\infty} \:\left\{\frac{\mathrm{1}^{{a}+\mathrm{1}} +\mathrm{2}^{{a}+\mathrm{1}} +...+{n}^{{a}+\mathrm{1}} }{{n}\left(\mathrm{1}^{{a}} +\mathrm{2}^{{a}} +....{n}^{{a}} \right)}\right\}=? \\ $$$$\:{where}\:{a}\:\neq−\mathrm{2}\:,\:−\mathrm{1} \\ $$

Question Number 124675    Answers: 0   Comments: 1

∫_1 ^3 (x−1)^3 (3−x)^2 dx

$$\int_{\mathrm{1}} ^{\mathrm{3}} \left({x}−\mathrm{1}\right)^{\mathrm{3}} \left(\mathrm{3}−{x}\right)^{\mathrm{2}} {dx} \\ $$

Question Number 124672    Answers: 1   Comments: 1

∫_(−3) ^(−2) (y+3)^6 (y+2)^4 dy

$$\int_{−\mathrm{3}} ^{−\mathrm{2}} \left({y}+\mathrm{3}\right)^{\mathrm{6}} \left({y}+\mathrm{2}\right)^{\mathrm{4}} {dy} \\ $$

Question Number 124654    Answers: 3   Comments: 1

∫_(1/(√2)) ^(1/2) (e^(cos^(−1) (x)) /( (√(1−x^2 )))) dx ?

$$\:\underset{\mathrm{1}/\sqrt{\mathrm{2}}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\frac{{e}^{\mathrm{cos}^{−\mathrm{1}} \left({x}\right)} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx}\:?\: \\ $$

Question Number 124644    Answers: 2   Comments: 0

∫_(2/(√3)) ^2 ((cos (sec^(−1) x))/(x(√(x^2 −1)))) dx ∫_( (√2)) ^2 ((sec^2 (sec^(−1) x))/(x(√(x^2 −1)))) dx

$$\underset{\mathrm{2}/\sqrt{\mathrm{3}}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{cos}\:\left(\mathrm{sec}^{−\mathrm{1}} {x}\right)}{{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:{dx}\: \\ $$$$\underset{\:\sqrt{\mathrm{2}}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{sec}\:^{\mathrm{2}} \left(\mathrm{sec}^{−\mathrm{1}} {x}\right)}{{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:{dx}\: \\ $$

Question Number 124632    Answers: 3   Comments: 0

Calculate ∫_0 ^2 (√((2+x)/(2−x))) dx

$${Calculate}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\sqrt{\frac{\mathrm{2}+{x}}{\mathrm{2}−{x}}}\:{dx}\: \\ $$

Question Number 124624    Answers: 1   Comments: 0

∫ (dx/( (x)^(1/3) +4x))

$$\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{{x}}\:+\mathrm{4}{x}} \\ $$

Question Number 124608    Answers: 2   Comments: 0

∫_0 ^∞ sinx^p dx ∫_0 ^∞ ((sinx^p )/x^q )dx collected question

$$\int_{\mathrm{0}} ^{\infty} {sinx}^{{p}} \:{dx} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}^{{p}} }{{x}^{{q}} }{dx} \\ $$$${collected}\:{question} \\ $$

Question Number 124607    Answers: 2   Comments: 0

∫(dx/((x^3 +1)^2 )) = ?

$$\int\frac{\boldsymbol{{dx}}}{\left(\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{2}} }\:=\:? \\ $$

Question Number 124594    Answers: 1   Comments: 1

Show that ∫_0 ^(ln 2) (1/(cosh(x + ln 4))) dx = 2 tan^(−1) ((4/(33)))

$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\mathrm{ln}\:\mathrm{2}} \frac{\mathrm{1}}{\mathrm{cosh}\left({x}\:+\:\mathrm{ln}\:\mathrm{4}\right)}\:{dx}\:=\:\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{4}}{\mathrm{33}}\right) \\ $$

Question Number 124587    Answers: 1   Comments: 0

...nice ◂::::▶ calculus simple question:: prove that :: ∫_0 ^( ∞) (4/( (√(4+x^4 )))) dx=^(???) ∫_0 ^( (π/2)) (dx/( (√(sin(x))))) +∫_0 ^( (π/2)) (dx/( (√(cos(x)))))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\blacktriangleleft::::\blacktriangleright\:{calculus} \\ $$$$\:\:\:\:\:{simple}\:\:{question}:: \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{4}}{\:\sqrt{\mathrm{4}+{x}^{\mathrm{4}} }}\:{dx}\overset{???} {=}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\:\sqrt{{sin}\left({x}\right)}}\:+\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\:\sqrt{{cos}\left({x}\right)}} \\ $$

Question Number 124579    Answers: 5   Comments: 0

I=∫_0 ^( ∞) (dx/((x+(√(x^2 +1)))^2 )) ?

$$\:\:{I}=\int_{\mathrm{0}} ^{\:\infty} \:\frac{{dx}}{\left({x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)^{\mathrm{2}} }\:? \\ $$

Question Number 124566    Answers: 4   Comments: 0

∫ (dx/((x+(√(x^2 +1)))^2 ))

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

Question Number 124542    Answers: 1   Comments: 1

help ∫3xdx

$${help}\:\:\int\mathrm{3}{xdx} \\ $$

Question Number 124540    Answers: 2   Comments: 0

∫((4x+9)/(x^2 +6x+10))dx

$$\int\frac{\mathrm{4}{x}+\mathrm{9}}{{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{10}}{dx} \\ $$

Question Number 124532    Answers: 5   Comments: 0

∫ (dx/(x^2 (√(x^2 −1)))) ?

$$\:\int\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:? \\ $$

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