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IntegrationQuestion and Answers: Page 150

Question Number 104421 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) ((ch(arctan(2x)))/(x^2 +x+1))dx

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

Question Number 104388 Answers: 1 Comments: 0

∫(dx/(x^2 ((x^3 −1))^(1/3) ))

$$\int\frac{\mathrm{d}{x}}{{x}^{\mathrm{2}} \sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} −\mathrm{1}}} \\ $$

Question Number 104359 Answers: 1 Comments: 0

solve this using Riemann sum f(x)=2x ; [0,4] for n=4

$${solve}\:{this}\:{using}\:{Riemann} \\ $$$${sum}\:{f}\left({x}\right)=\mathrm{2}{x}\:;\:\left[\mathrm{0},\mathrm{4}\right]\:{for}\:{n}=\mathrm{4} \\ $$

Question Number 104851 Answers: 4 Comments: 0

∫ ((√(x^2 −9))/x^3 ) dx

$$\int\:\frac{\sqrt{{x}^{\mathrm{2}} −\mathrm{9}}}{{x}^{\mathrm{3}} }\:{dx}\: \\ $$

Question Number 104339 Answers: 3 Comments: 1

Examine ∫_0 ^3 ((2x)/((1−x^2 )^(2/3) )) dx

$${Examine}\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\frac{\mathrm{2}{x}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}/\mathrm{3}} }\:{dx} \\ $$

Question Number 104338 Answers: 0 Comments: 0

∫((tdt)/((1+t^3 )((1+t^3 ))^(1/3) ))

$$\int\frac{\mathrm{tdt}}{\left(\mathrm{1}+\mathrm{t}^{\mathrm{3}} \right)\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{t}^{\mathrm{3}} }} \\ $$

Question Number 104312 Answers: 1 Comments: 0

∫_0 ^(π/4) ((√(sin^2 θ+2))/(sinθ))dθ

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\sqrt{\mathrm{sin}^{\mathrm{2}} \theta+\mathrm{2}}}{\mathrm{sin}\theta}\mathrm{d}\theta \\ $$

Question Number 104264 Answers: 2 Comments: 0

∫_0 ^( (√2)) ∫_y ^( (√(4−y^2 ))) (1/(√(1+x^2 +y^2 )))dxdy

$$\int_{\mathrm{0}} ^{\:\sqrt{\mathrm{2}}} \:\int_{{y}} ^{\:\sqrt{\mathrm{4}−{y}^{\mathrm{2}} }} \frac{\mathrm{1}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }}{dxdy} \\ $$

Question Number 104220 Answers: 3 Comments: 1

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

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

Question Number 104197 Answers: 2 Comments: 0

calculate ∫_5 ^(+∞) (dx/((x^2 −9)^4 ))

$$\mathrm{calculate}\:\int_{\mathrm{5}} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{9}\right)^{\mathrm{4}} } \\ $$

Question Number 104180 Answers: 0 Comments: 0

Question Number 104104 Answers: 4 Comments: 0

∫ ((xtan^(−1) (x))/(√(1+x^2 ))) dx ?

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

Question Number 104028 Answers: 2 Comments: 0

calculate ∫_(20) ^(+∞) (dx/((x−18)^(19) (x−19)^(18) ))

$$\mathrm{calculate}\:\:\int_{\mathrm{20}} ^{+\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}−\mathrm{18}\right)^{\mathrm{19}} \left(\mathrm{x}−\mathrm{19}\right)^{\mathrm{18}} } \\ $$

Question Number 103974 Answers: 2 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)(x^2 +3)^2 ))

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

Question Number 103871 Answers: 1 Comments: 2

∫_0 ^1 ((x^(98) −99x+98)/(logx))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{98}} −\mathrm{99}{x}+\mathrm{98}}{{logx}}{dx} \\ $$

Question Number 103863 Answers: 8 Comments: 0

Question Number 103860 Answers: 2 Comments: 0

find ∫ (dx/(cos^4 x))

$$\mathrm{find}\:\int\:\frac{\mathrm{dx}}{\mathrm{cos}^{\mathrm{4}} \mathrm{x}} \\ $$

Question Number 103846 Answers: 0 Comments: 2

∫(dx/(x^(1/3) +2))

$$\int\frac{{dx}}{{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{2}} \\ $$

Question Number 103825 Answers: 7 Comments: 0

∫ (dx/((1−sinx)^2 )) ?

$$\int\:\frac{{dx}}{\left(\mathrm{1}−{sinx}\right)^{\mathrm{2}} }\:? \\ $$

Question Number 103773 Answers: 1 Comments: 0

∫_c ((x^2 +2xy^2 )dx+(x^2 y^2 −1)dy) where C is the boundary of region define by y^2 = 4x and y =1 ?

$$\int_{{c}} \left(\left({x}^{\mathrm{2}} +\mathrm{2}{xy}^{\mathrm{2}} \right){dx}+\left({x}^{\mathrm{2}} {y}^{\mathrm{2}} −\mathrm{1}\right){dy}\right) \\ $$$${where}\:{C}\:{is}\:{the}\:{boundary}\:{of} \\ $$$${region}\:{define}\:{by}\:{y}^{\mathrm{2}} =\:\mathrm{4}{x}\:{and}\:{y} \\ $$$$=\mathrm{1}\:? \\ $$

Question Number 103742 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/((2x+1)^4 (x+3)^5 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\mathrm{dx}}{\left(\mathrm{2x}+\mathrm{1}\right)^{\mathrm{4}} \left(\mathrm{x}+\mathrm{3}\right)^{\mathrm{5}} } \\ $$

Question Number 103741 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((arctan(ch(x)))/(x^2 +9))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{arctan}\left(\mathrm{ch}\left(\mathrm{x}\right)\right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{9}}\mathrm{dx} \\ $$

Question Number 103723 Answers: 0 Comments: 0

Solve for n, such that; 1−(1/2)+∙∙∙+(((−1)^n )/(n+1))=∫_0 ^1 (x^(n+1) /(1+x))dx−ln2−(−1)^(n+1)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{n},\:\mathrm{such}\:\mathrm{that}; \\ $$$$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\centerdot\centerdot\centerdot+\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}+\mathrm{1}}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\mathrm{1}+\mathrm{x}}\mathrm{dx}−\mathrm{ln2}−\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{1}} \\ $$

Question Number 103683 Answers: 1 Comments: 1

∫_0 ^1 tan^(−1) (((2x−1)/(1+x−x^2 )))dx

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

Question Number 103607 Answers: 1 Comments: 0

what is the value of ∫_c (x+2y)dx+(4−2x)dy around the ellipse C: (x^2 /(16))+(y^2 /8)=1 in the counterclockwise direction ?

$${what}\:{is}\:{the}\:{value}\:{of}\:\int_{{c}} \left({x}+\mathrm{2}{y}\right){dx}+\left(\mathrm{4}−\mathrm{2}{x}\right){dy} \\ $$$${around}\:{the}\:{ellipse}\:{C}:\:\frac{{x}^{\mathrm{2}} }{\mathrm{16}}+\frac{{y}^{\mathrm{2}} }{\mathrm{8}}=\mathrm{1} \\ $$$${in}\:{the}\:{counterclockwise} \\ $$$${direction}\:?\: \\ $$

Question Number 103593 Answers: 1 Comments: 0

calculate ∫_(−∞) ^∞ (dx/((x^2 +x +1)^2 (2x^2 +5)^2 ))

$$\mathrm{calculate}\:\:\int_{−\infty} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}\:+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2x}^{\mathrm{2}} \:+\mathrm{5}\right)^{\mathrm{2}} } \\ $$

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