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IntegrationQuestion and Answers: Page 199

Question Number 82971 Answers: 1 Comments: 2

1)find ∫ (dx/((x^2 +x+1)^6 )) 2)calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^6 ))

$$\left.\mathrm{1}\right){find}\:\int\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{6}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{6}} } \\ $$

Question Number 82970 Answers: 0 Comments: 2

1)find ∫ (dx/((x^2 −1)^9 )) 2) calculate ∫_2 ^(+∞) (dx/((x^2 −1)^9 ))

$$\left.\mathrm{1}\right){find}\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{9}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{2}} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{9}} } \\ $$

Question Number 82954 Answers: 1 Comments: 3

∫ ((cos 4x−cos 2x)/(sin 4x−cos 2x)) dx

$$\int\:\frac{\mathrm{cos}\:\mathrm{4x}−\mathrm{cos}\:\mathrm{2x}}{\mathrm{sin}\:\mathrm{4x}−\mathrm{cos}\:\mathrm{2x}}\:\mathrm{dx}\: \\ $$

Question Number 82891 Answers: 2 Comments: 0

∫ (1/(√(1−x^4 ))) dx = ?

$$\int\:\frac{\mathrm{1}}{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}\:\mathrm{dx}\:=\:? \\ $$

Question Number 82872 Answers: 0 Comments: 1

1)find ∫∫_W ((xdx)/(a^2 +x^2 +y^2 )) with W_a →x^2 +y^2 ≤a^2 and x>0 (a>0) 2)calculate ∫∫_W_1 ((xdx)/(x^2 +y^2 +1))

$$\left.\mathrm{1}\right){find}\:\int\int_{{W}} \:\frac{{xdx}}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:{with} \\ $$$${W}_{{a}} \rightarrow{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant{a}^{\mathrm{2}} \:{and}\:{x}>\mathrm{0}\:\:\:\:\:\left({a}>\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int\int_{{W}_{\mathrm{1}} } \:\:\:\frac{{xdx}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

Question Number 82871 Answers: 0 Comments: 1

find ∫∫_([0,1]^2 ) ∣x^2 −y^2 ∣dxdy

$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\mid{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \mid{dxdy} \\ $$

Question Number 82816 Answers: 0 Comments: 3

calculate the exact value ∫_0 ^∞ ((cos(x))/(x^2 +1)) dx

$${calculate}\:{the}\:{exact}\:{value}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{cos}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx} \\ $$

Question Number 82755 Answers: 1 Comments: 2

calculate ∫_0 ^∞ ((lnx)/((1+x^2 )^2 ))dx

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

Question Number 82700 Answers: 1 Comments: 4

∫ ((2dx)/(3x(√(5x^2 +6)))) ?

$$\int\:\frac{\mathrm{2}{dx}}{\mathrm{3}{x}\sqrt{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{6}}}\:? \\ $$

Question Number 82617 Answers: 1 Comments: 0

∫sin (101x)(sinx)^(99) dx

$$\int\mathrm{sin}\:\left(\mathrm{101}{x}\right)\left({sinx}\right)^{\mathrm{99}} {dx} \\ $$

Question Number 82614 Answers: 1 Comments: 0

∫ ((sin 2x)/(sin 5x sin 3x)) dx ?

$$\int\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{sin}\:\mathrm{5}{x}\:\mathrm{sin}\:\mathrm{3}{x}}\:{dx}\:?\: \\ $$

Question Number 82685 Answers: 0 Comments: 0

∫_0 ^5 (((3x^3 −x^4 ))^(1/4) /(5−x))dx

$$\int_{\mathrm{0}} ^{\mathrm{5}} \:\frac{\sqrt[{\mathrm{4}}]{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{4}} }}{\mathrm{5}−{x}}{dx} \\ $$

Question Number 82604 Answers: 1 Comments: 0

∫ ((cos (5x)+cos (4x) dx)/(1−2cos (3x))) =

$$\int\:\frac{\mathrm{cos}\:\left(\mathrm{5}{x}\right)+\mathrm{cos}\:\left(\mathrm{4}{x}\right)\:{dx}}{\mathrm{1}−\mathrm{2cos}\:\left(\mathrm{3}{x}\right)}\:=\: \\ $$

Question Number 82577 Answers: 0 Comments: 3

∫((2−(√(x+3)))/(2+(√(x−3)))) dx

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

Question Number 82573 Answers: 0 Comments: 4

Question Number 82571 Answers: 0 Comments: 0

Question Number 82568 Answers: 1 Comments: 0

∫ (1/(tan x+cot x+sec x+cosec x)) dx ?

$$\int\:\frac{\mathrm{1}}{\mathrm{tan}\:{x}+\mathrm{cot}\:{x}+\mathrm{sec}\:{x}+\mathrm{cosec}\:{x}}\:{dx}\:? \\ $$

Question Number 82566 Answers: 2 Comments: 1

∫ (√((x+1)/x)) dx = ?

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

Question Number 82560 Answers: 1 Comments: 0

Use gamma function to prove (i) ∫_0 ^(π/4) sin^4 x 2x dx = ((3𝛑−4)/(192)). (ii) ∫_0 ^(π/6) cos^4 3𝛉 sin^2 6θ = ((5π)/(192)).

$$\:\boldsymbol{\mathrm{U}}\mathrm{se}\:\mathrm{gamma}\:\mathrm{function}\:\mathrm{to}\:\mathrm{prove} \\ $$$$\:\:\left(\mathrm{i}\right)\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \boldsymbol{\mathrm{sin}}^{\mathrm{4}} \boldsymbol{\mathrm{x}}\:\mathrm{2}\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{dx}}\:=\:\frac{\mathrm{3}\boldsymbol{\pi}−\mathrm{4}}{\mathrm{192}}. \\ $$$$\:\:\left(\mathrm{ii}\right)\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\boldsymbol{\mathrm{cos}}^{\mathrm{4}} \mathrm{3}\boldsymbol{\theta}\:\mathrm{sin}^{\mathrm{2}} \mathrm{6}\theta\:=\:\frac{\mathrm{5}\pi}{\mathrm{192}}. \\ $$

Question Number 82531 Answers: 1 Comments: 1

Question Number 82510 Answers: 1 Comments: 1

∫ (√(x+(√x) )) dx = ?

$$\int\:\sqrt{{x}+\sqrt{{x}}\:}\:{dx}\:=\:? \\ $$

Question Number 82450 Answers: 1 Comments: 1

∫_0 ^π (1/(1+(tan(x))^(√2) )) dx

$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{1}}{\mathrm{1}+\left({tan}\left({x}\right)\right)^{\sqrt{\mathrm{2}}} }\:{dx} \\ $$

Question Number 82446 Answers: 0 Comments: 3

show that ∫_0 ^∞ x^(−log(x)) x log(x) dx=e(√π)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{−{log}\left({x}\right)} \:{x}\:{log}\left({x}\right)\:{dx}={e}\sqrt{\pi} \\ $$

Question Number 82440 Answers: 0 Comments: 1

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

$${find}\:\int\:\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx} \\ $$

Question Number 82439 Answers: 0 Comments: 1

calculate I_n =∫_0 ^1 x^n (√(1+x+x^2 ))dx

$${calculate}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}+{x}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 82442 Answers: 0 Comments: 1

1)find ∫ ((√(x^2 −x+1))/(x^2 +3))dx 2)calculate ∫_0 ^1 ((√(x^2 −x+1))/(x^2 +3))dx

$$\left.\mathrm{1}\right){find}\:\int\:\frac{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

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