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IntegrationQuestion and Answers: Page 51

Question Number 167355 Answers: 0 Comments: 4

Question Number 167328 Answers: 1 Comments: 0

∫_0 ^( π/2) ((cos x+cos^5 x sin x)/( (√(1+cos^2 x)))) dx =?

$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \:\frac{\mathrm{cos}\:{x}+\mathrm{cos}\:^{\mathrm{5}} {x}\:\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}}\:{dx}\:=? \\ $$

Question Number 167318 Answers: 1 Comments: 0

∫ (dx/( (√(x+(x)^(1/3) )))) =?

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

Question Number 167312 Answers: 2 Comments: 1

∫ ((sin^3 x+1)/(sin^2 x−1)) dx =?

$$\:\:\:\:\:\:\:\int\:\frac{\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{1}}\:\mathrm{dx}\:=? \\ $$

Question Number 167310 Answers: 0 Comments: 0

NICE CALCULUS ∫_0 ^1 ((ln(1−x)ln(1−(x/2)))/x)dx=?

$$\boldsymbol{\mathrm{NICE}}\:\boldsymbol{\mathrm{CALCULUS}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}}=? \\ $$$$ \\ $$$$ \\ $$

Question Number 167305 Answers: 1 Comments: 0

Q=∫ ((2sin (x))/( (√3) sin (x)−cos (x))) dx=?

$$\:\:\:{Q}=\int\:\frac{\mathrm{2sin}\:\left({x}\right)}{\:\sqrt{\mathrm{3}}\:\mathrm{sin}\:\left({x}\right)−\mathrm{cos}\:\left({x}\right)}\:{dx}=? \\ $$

Question Number 170532 Answers: 1 Comments: 0

Let I_n =∫x^n e^(−x) dx, n = 0,1,2,... (i) Show that I_n = −x^n e^(−x) +nI_(n−1) (ii) Show that ∫_0 ^∞ x^n e^(−x) dx = n!

$$\mathrm{Let}\:{I}_{{n}} \:=\int{x}^{{n}} {e}^{−{x}} {dx},\:{n}\:=\:\mathrm{0},\mathrm{1},\mathrm{2},... \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:{I}_{{n}} \:=\:−{x}^{{n}} {e}^{−{x}} +{nI}_{{n}−\mathrm{1}} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} {x}^{{n}} {e}^{−{x}} {dx}\:=\:{n}! \\ $$

Question Number 167223 Answers: 0 Comments: 3

∫ sin^3 (3x) cos^4 (5x) dx=?

$$\:\:\int\:\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{3x}\right)\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{5x}\right)\:\mathrm{dx}=? \\ $$

Question Number 167213 Answers: 1 Comments: 0

solve I= ∫_0 ^( (1/2)) ((ln^( 2) (x))/(1−x)) dx =?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{solve} \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\mathcal{I}=\:\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{{ln}^{\:\mathrm{2}} \left({x}\right)}{\mathrm{1}−{x}}\:{dx}\:=? \\ $$$$ \\ $$

Question Number 167173 Answers: 1 Comments: 0

∫_(−2) ^(−1) e^(−(t/2)) (√(t+2)) dt = ???

$$\int_{−\mathrm{2}} ^{−\mathrm{1}} {e}^{−\frac{{t}}{\mathrm{2}}} \sqrt{{t}+\mathrm{2}}\:{dt}\:=\:??? \\ $$

Question Number 167150 Answers: 1 Comments: 0

Question Number 167115 Answers: 0 Comments: 0

Question Number 167102 Answers: 0 Comments: 0

Ω = ∫_0 ^( 1) ((Li_( 2) (1− x ))/(1+x)) dx = ? −−−−−

$$ \\ $$$$\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{Li}_{\:\mathrm{2}} \left(\mathrm{1}−\:{x}\:\right)}{\mathrm{1}+{x}}\:{dx}\:=\:? \\ $$$$\:\:\:\:\:\:−−−−− \\ $$

Question Number 167092 Answers: 0 Comments: 0

Question Number 167082 Answers: 1 Comments: 1

Question Number 167048 Answers: 1 Comments: 0

prove that Φ= ∫_0 ^( 1) x.ψ (2+x )= 2 −(1/2)ln(8π) −−−

$$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\Phi=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}.\psi\:\left(\mathrm{2}+{x}\:\right)=\:\mathrm{2}\:−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{8}\pi\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:−−− \\ $$

Question Number 167040 Answers: 0 Comments: 1

∫_0 ^( (π/2)) (1/(1+sin^6 x)) dx=?

$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}}\:\mathrm{dx}=? \\ $$

Question Number 167025 Answers: 0 Comments: 0

∫_0 ^(π/3) (√(1−(1/(3 ))sin^2 θ)) dθ

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \sqrt{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}\:}\mathrm{sin}\:^{\mathrm{2}} \theta}\:\mathrm{d}\theta \\ $$

Question Number 167024 Answers: 1 Comments: 0

∫sec θtan^4 θdθ

$$\int\mathrm{sec}\:\theta\mathrm{tan}\:^{\mathrm{4}} \theta\mathrm{d}\theta \\ $$

Question Number 167006 Answers: 1 Comments: 0

∫ ((3x^3 )/((x−1)^3 )) dx=?

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

Question Number 166959 Answers: 1 Comments: 0

γ=∫ ((e^x (sin x+1))/(cos x+1)) dx =?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\gamma=\int\:\frac{\mathrm{e}^{\mathrm{x}} \left(\mathrm{sin}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{cos}\:\mathrm{x}+\mathrm{1}}\:\mathrm{dx}\:=? \\ $$

Question Number 166956 Answers: 1 Comments: 0

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

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

Question Number 166940 Answers: 1 Comments: 0

Question Number 166939 Answers: 1 Comments: 0

Question Number 166916 Answers: 0 Comments: 0

Ω= Σ_(n=1) ^∞ (( H_( n) )/(n(n+1))) = −−−−−− Ω = Σ_(n=1) ^∞ −(1/(n+1)) ∫_(0 ) ^( 1) x^( n−1) ln(1−x )dx = ∫_0 ^( 1) {−(1/x^2 )ln(1−x).Σ_(n=1) (x^( n+1) /(n+1))}dx = ∫_0 ^( 1) {((−ln(1−x))/x^( 2) )Σ_(n=2) ^∞ (x^( n) /n)}dx = ∫_0 ^( 1) ((−ln(1−x))/x^( 2) ) {−x +Σ_(n=1) ^∞ (x^( n) /n) }dx = −li_( 2) ( 1) +[ ∫_0 ^( 1) ((ln^( 2) ( 1−x ))/x^( 2) )dx=_(derived) ^(earlier) (π^( 2) /3) ] = −(π^( 2) /6) + (π^( 2) /3) = (( π^( 2) )/6) = ζ (2) ■ m.n

$$ \\ $$$$\:\:\:\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{H}_{\:{n}} }{{n}\left({n}+\mathrm{1}\right)}\:= \\ $$$$\:\:\:\:\:\:\:−−−−−− \\ $$$$\:\:\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\int_{\mathrm{0}\:} ^{\:\mathrm{1}} {x}^{\:{n}−\mathrm{1}} {ln}\left(\mathrm{1}−{x}\:\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{ln}\left(\mathrm{1}−{x}\right).\underset{{n}=\mathrm{1}} {\sum}\frac{{x}^{\:{n}+\mathrm{1}} }{{n}+\mathrm{1}}\right\}{dx} \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{\frac{−{ln}\left(\mathrm{1}−{x}\right)}{{x}^{\:\mathrm{2}} }\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{{x}^{\:{n}} }{{n}}\right\}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{−{ln}\left(\mathrm{1}−{x}\right)}{{x}^{\:\mathrm{2}} }\:\left\{−{x}\:+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{\:{n}} }{{n}}\:\right\}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:−{li}_{\:\mathrm{2}} \left(\:\mathrm{1}\right)\:+\left[\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\:\mathrm{2}} \left(\:\mathrm{1}−{x}\:\right)}{{x}^{\:\mathrm{2}} }{dx}\underset{{derived}} {\overset{{earlier}} {=}}\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:−\frac{\pi^{\:\mathrm{2}} }{\mathrm{6}}\:+\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:=\:\frac{\:\pi^{\:\mathrm{2}} }{\mathrm{6}}\:=\:\zeta\:\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$

Question Number 166829 Answers: 1 Comments: 0

calculate If , f(x)=(( (x^2 +1)(((x^( 2) +x−2)(x^( 4) −1)(x^( 2) +2x−3)+16))^(1/3) + (√(x^( 2) +3)))/(( 1+x +x^( 2) ))) then , f ′ (1 ) =? ■ m.n

$$ \\ $$$$\:\:\:\:\:\:\:\:{calculate}\: \\ $$$$\:\:\:\mathrm{I}{f}\:,\:\:\:\:{f}\left({x}\right)=\frac{\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt[{\mathrm{3}}]{\left({x}^{\:\mathrm{2}} +{x}−\mathrm{2}\right)\left({x}^{\:\mathrm{4}} −\mathrm{1}\right)\left({x}^{\:\mathrm{2}} +\mathrm{2}{x}−\mathrm{3}\right)+\mathrm{16}}\:\:+\:\sqrt{{x}^{\:\mathrm{2}} +\mathrm{3}}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} \right)} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{then}\:,\:\:\:\:{f}\:'\:\left(\mathrm{1}\:\right)\:=?\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$ \\ $$

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