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IntegrationQuestion and Answers: Page 147

Question Number 113868 Answers: 0 Comments: 1

Find the area between the circle ρ=2acosθ and cardiode ρ=a(1+cosθ)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{between}\:\mathrm{the}\:\mathrm{circle}\:\rho=\mathrm{2acos}\theta\:\mathrm{and}\: \\ $$$$\mathrm{cardiode}\:\rho=\mathrm{a}\left(\mathrm{1}+\mathrm{cos}\theta\right) \\ $$

Question Number 113867 Answers: 1 Comments: 0

∫_3 ^6 ((x+1)/(x^3 +x^2 −6x))dx

$$\int_{\mathrm{3}} ^{\mathrm{6}} \frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} −\mathrm{6x}}\mathrm{dx} \\ $$

Question Number 113865 Answers: 0 Comments: 0

Consider the series I_n =∫_1 ^e x(lnx)^n dx and I_0 =∫_1 ^e xdx Which of the following is true ? a\ 0≤I_n ≤(e^2 /(n+2)) b\1≤I_n ≤(e^2 /(n+1)) c\I_n is negative

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{series}\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{1}} ^{\mathrm{e}} \mathrm{x}\left(\mathrm{lnx}\right)^{\mathrm{n}} \mathrm{dx}\:\mathrm{and}\:\mathrm{I}_{\mathrm{0}} =\int_{\mathrm{1}} ^{\mathrm{e}} \mathrm{xdx} \\ $$$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{true}\:? \\ $$$$\mathrm{a}\backslash\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\frac{\mathrm{e}^{\mathrm{2}} }{\mathrm{n}+\mathrm{2}}\:\:\:\:\mathrm{b}\backslash\mathrm{1}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\frac{\mathrm{e}^{\mathrm{2}} }{\mathrm{n}+\mathrm{1}}\:\:\mathrm{c}\backslash\mathrm{I}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{negative} \\ $$

Question Number 113821 Answers: 3 Comments: 1

∫_0 ^(π/2) ln(2−sinx)dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{2}−\mathrm{sinx}\right)\mathrm{dx} \\ $$

Question Number 113766 Answers: 1 Comments: 0

I=∫_0 ^∞ ((π/(1+π^2 x^2 ))−(1/(1+x^2 )))lnx dx put πx=tanA, x =tanB I=∫_0 ^(π/2) (ln(tanA)−lnπ)dA−∫_0 ^(π/2) ln(tanB)dB I=((−π)/2)lnπ

$$ \\ $$$${I}=\int_{\mathrm{0}} ^{\infty} \left(\frac{\pi}{\mathrm{1}+\pi^{\mathrm{2}} {x}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }\right){lnx}\:{dx} \\ $$$${put}\:\pi{x}={tanA},\:{x}\:={tanB} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({ln}\left({tanA}\right)−{ln}\pi\right){dA}−\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {ln}\left({tanB}\right){dB} \\ $$$${I}=\frac{−\pi}{\mathrm{2}}{ln}\pi \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 113760 Answers: 1 Comments: 0

∫(√((x−1)/x^5 ))dx

$$\int\sqrt{\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}^{\mathrm{5}} }}\mathrm{dx} \\ $$

Question Number 113745 Answers: 1 Comments: 0

Question Number 113738 Answers: 1 Comments: 0

∫_0 ^π ((x sin x)/(1+cos^2 x)) dx ?

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

Question Number 113757 Answers: 2 Comments: 0

∫ (dx/(tan x−sin x)) ?

$$\:\int\:\frac{{dx}}{\mathrm{tan}\:{x}−\mathrm{sin}\:{x}}\:?\: \\ $$

Question Number 113675 Answers: 1 Comments: 3

Question Number 113656 Answers: 1 Comments: 0

∫ (((1+tan (((3x)/2)))^2 )/(1+sin 3x)) dx ?

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

Question Number 113634 Answers: 2 Comments: 0

Bonjour besoin d′aide Calculer ∫ln(cosx)dx

$${Bonjour}\:{besoin}\:{d}'{aide} \\ $$$${Calculer}\:\int{ln}\left({cosx}\right){dx} \\ $$

Question Number 113630 Answers: 3 Comments: 0

explicit g(a) =∫_0 ^(π/4) ln(1+acos^2 θ)dθ

$$\mathrm{explicit}\:\mathrm{g}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\mathrm{ln}\left(\mathrm{1}+\mathrm{acos}^{\mathrm{2}} \theta\right)\mathrm{d}\theta \\ $$

Question Number 113629 Answers: 0 Comments: 0

find f(a) =∫_0 ^(π/8) ln(1+a sinθ)dθ with o<a<1

$$\mathrm{find}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:\mathrm{ln}\left(\mathrm{1}+\mathrm{a}\:\mathrm{sin}\theta\right)\mathrm{d}\theta\:\:\:\mathrm{with}\:\mathrm{o}<\mathrm{a}<\mathrm{1} \\ $$

Question Number 113600 Answers: 1 Comments: 0

Prouver que β(a,b)=((Γ(a)Γ(b))/(Γ(a+b)))=∫_0 ^1 x^(a−1) (1−x)^(b−1) dx

$${Prouver}\:{que} \\ $$$$\beta\left({a},{b}\right)=\frac{\Gamma\left({a}\right)\Gamma\left({b}\right)}{\Gamma\left({a}+{b}\right)}=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{a}−\mathrm{1}} \left(\mathrm{1}−{x}\right)^{{b}−\mathrm{1}} {dx} \\ $$

Question Number 113628 Answers: 1 Comments: 0

find ∫ (dx/((x+1)(√(x^2 −1))+(x−1)(√(x^2 +1))))

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

Question Number 113627 Answers: 0 Comments: 0

calculate ∫_0 ^∞ (dx/(x^4 +ix^2 +2))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{4}} +\mathrm{ix}^{\mathrm{2}} \:+\mathrm{2}} \\ $$

Question Number 113565 Answers: 1 Comments: 0

Question Number 113564 Answers: 1 Comments: 0

∫_1 ^2 f(2x+1) dx = 3 and ∫_1 ^2 (x^2 f(x^3 +2))dx = 5 then ∫_5 ^(10) f(x) dx = ?

$$\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {f}\left(\mathrm{2}{x}+\mathrm{1}\right)\:{dx}\:=\:\mathrm{3}\:{and}\: \\ $$$$\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \left({x}^{\mathrm{2}} \:{f}\left({x}^{\mathrm{3}} +\mathrm{2}\right)\right){dx}\:=\:\mathrm{5} \\ $$$${then}\:\int_{\mathrm{5}} ^{\mathrm{10}} {f}\left({x}\right)\:{dx}\:=\:? \\ $$

Question Number 113562 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/(x^4 +ix^2 +2))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{4}} +\mathrm{ix}^{\mathrm{2}} \:+\mathrm{2}} \\ $$

Question Number 113421 Answers: 0 Comments: 2

Question Number 113418 Answers: 3 Comments: 1

∫ ((3x−2)/( (√(x^2 +2x+26)))) dx

$$\:\int\:\frac{\mathrm{3x}−\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}\:\mathrm{dx} \\ $$

Question Number 113323 Answers: 2 Comments: 0

Question Number 113275 Answers: 2 Comments: 0

∫ (dx/(3sin x+sin^3 x)) ?

$$\:\int\:\frac{\mathrm{dx}}{\mathrm{3sin}\:\mathrm{x}+\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}}\:? \\ $$

Question Number 113218 Answers: 1 Comments: 0

Question Number 113203 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/(x^4 +2x^2 +3))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}} \\ $$

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