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IntegrationQuestion and Answers: Page 167

Question Number 97800 Answers: 3 Comments: 0

1) findf(a)= ∫_0 ^1 (√(x^2 −x+a))dx with a>(1/2) 2)explicite g(a) =∫_0 ^1 (dx/(√(x^2 −x+a))) 3) calculate ∫_0 ^1 (dx/(√(x^2 −x +3)))

$$\left.\mathrm{1}\right)\:\mathrm{findf}\left(\mathrm{a}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{a}}\mathrm{dx}\:\:\:\:\:\mathrm{with}\:\mathrm{a}>\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\mathrm{explicite}\:\mathrm{g}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{a}}}\: \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}\:+\mathrm{3}}} \\ $$

Question Number 97794 Answers: 1 Comments: 3

solve y^(′′) +y =(1/(cosx))

$$\mathrm{solve}\:\mathrm{y}^{''} \:+\mathrm{y}\:=\frac{\mathrm{1}}{\mathrm{cosx}} \\ $$

Question Number 97782 Answers: 2 Comments: 1

Evaluate: ∫ ((sinx)/(1 +sin^2 x))dx

$${Evaluate}: \\ $$$$\int\:\frac{{sinx}}{\mathrm{1}\:+{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 97759 Answers: 1 Comments: 0

∫ ((sin^5 (x) dx)/(√(cos (x)))) ?

$$\int\:\frac{\mathrm{sin}\:^{\mathrm{5}} \left({x}\right)\:{dx}}{\sqrt{\mathrm{cos}\:\left({x}\right)}}\:? \\ $$

Question Number 97721 Answers: 1 Comments: 3

Question Number 97892 Answers: 1 Comments: 0

Question Number 97707 Answers: 3 Comments: 0

Question Number 97683 Answers: 3 Comments: 3

Evaluate ∫_0 ^1 (1/(√(16 + 9x^2 ))) dx

$$\:\mathrm{Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\sqrt{\mathrm{16}\:+\:\mathrm{9}{x}^{\mathrm{2}} }}\:{dx} \\ $$

Question Number 97671 Answers: 0 Comments: 1

Question Number 97660 Answers: 2 Comments: 1

Question Number 97638 Answers: 0 Comments: 0

Question Number 97627 Answers: 2 Comments: 0

give ∫_0 ^∞ ((arctan(x))/((1+x^2 )^2 ))dx at form of serie

$$\mathrm{give}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dx}\:\mathrm{at}\:\mathrm{form}\:\mathrm{of}\:\mathrm{serie} \\ $$

Question Number 97625 Answers: 1 Comments: 0

solve x^2 y^(′′) −(x+1)y^′ =x^2 sinx

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

Question Number 97624 Answers: 2 Comments: 0

calculate ∫_2 ^∞ (dx/((x+1)^3 (x^2 +1)^4 ))

$$\mathrm{calculate}\:\int_{\mathrm{2}} ^{\infty} \:\:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} \left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 97620 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((sin(πx^2 ))/(x^4 −x^2 +1))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{sin}\left(\pi\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{4}} −\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$

Question Number 97619 Answers: 1 Comments: 0

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

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

Question Number 97591 Answers: 1 Comments: 0

∫(x^4 /(x^3 −2x^2 −7x+4))dx

$$\int\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} −\mathrm{7}{x}+\mathrm{4}}{dx} \\ $$

Question Number 97569 Answers: 1 Comments: 0

∫_0 ^(π/4) (√(tan(x)))(√(1−tan(x))) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \sqrt{{tan}\left({x}\right)}\sqrt{\mathrm{1}−{tan}\left({x}\right)}\:{dx} \\ $$

Question Number 97552 Answers: 2 Comments: 0

∫_0 ^1 (√(x/(1−x))) ln((x/(1−x))) dx ?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt{\frac{{x}}{\mathrm{1}−{x}}}\:\mathrm{ln}\left(\frac{{x}}{\mathrm{1}−{x}}\right)\:{dx}\:? \\ $$

Question Number 97550 Answers: 0 Comments: 0

Question Number 97537 Answers: 1 Comments: 0

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

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{\left({yx}^{\mathrm{3}} +{x}^{\mathrm{2}} −{yx}−\mathrm{1}\right)}{dx} \\ $$

Question Number 97531 Answers: 1 Comments: 0

5050((∫_0 ^1 (1−x^(50) )^(100) dx)/(∫_0 ^1 (1−x^(50) )^(101) dx))=

$$\mathrm{5050}\frac{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{50}} \right)^{\mathrm{100}} {dx}}{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{50}} \right)^{\mathrm{101}} {dx}}= \\ $$

Question Number 97527 Answers: 0 Comments: 0

∫((xdx)/(x!))=?

$$\int\frac{\mathrm{xdx}}{\mathrm{x}!}=? \\ $$

Question Number 97526 Answers: 1 Comments: 1

∫(dx/(1+sin x))=?

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

Question Number 97505 Answers: 0 Comments: 0

Question Number 97439 Answers: 0 Comments: 2

∫_((√2)/2) ^1 ((x^3 /2) + (1/(6x)))(√(1+(((3x^2 )/2) −(1/(6x^2 )))^2 )) dx

$$\underset{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}} {\overset{\mathrm{1}} {\int}}\:\left(\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{6x}}\right)\sqrt{\mathrm{1}+\left(\frac{\mathrm{3x}^{\mathrm{2}} }{\mathrm{2}}\:−\frac{\mathrm{1}}{\mathrm{6x}^{\mathrm{2}} }\right)^{\mathrm{2}} }\:\:\mathrm{dx} \\ $$

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