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IntegrationQuestion and Answers: Page 186

Question Number 88438 Answers: 2 Comments: 0

∫((x^5 +1)/(x^5 −1))dx

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

Question Number 88429 Answers: 0 Comments: 0

show that ∫_(0 ) ^1 ln(x) sin^(−1) (√x) dx= (π/2)(ln(2)−1)

$${show}\:{that} \\ $$$$\int_{\mathrm{0}\:} ^{\mathrm{1}} {ln}\left({x}\right)\:{sin}^{−\mathrm{1}} \sqrt{{x}}\:{dx}=\:\frac{\pi}{\mathrm{2}}\left({ln}\left(\mathrm{2}\right)−\mathrm{1}\right) \\ $$

Question Number 88422 Answers: 0 Comments: 1

calculate ∫_1 ^∞ (dx/((x+1)^3 (x^2 +1)^2 ))

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

Question Number 88415 Answers: 0 Comments: 2

find L(((1−cosx)/x^2 )) with L lsplace transform

$${find}\:{L}\left(\frac{\mathrm{1}−{cosx}}{{x}^{\mathrm{2}} }\right)\:{with}\:{L}\:{lsplace}\:{transform} \\ $$

Question Number 88414 Answers: 0 Comments: 2

find approcimstive value of ∫_(π/3) ^(π/2) (x/(sinx))dx

$${find}\:{approcimstive}\:{value}\:{of}\:\:\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{x}}{{sinx}}{dx} \\ $$

Question Number 88413 Answers: 0 Comments: 3

∫_0 ^∞ e^(−x^2 ) dx

$$\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$$$ \\ $$

Question Number 88388 Answers: 1 Comments: 0

Question Number 88357 Answers: 0 Comments: 1

∫ _1 ^4 (dx/((4x−1)(√x)))

$$\int\underset{\mathrm{1}} {\overset{\mathrm{4}} {\:}}\:\frac{\mathrm{dx}}{\left(\mathrm{4x}−\mathrm{1}\right)\sqrt{\mathrm{x}}} \\ $$

Question Number 88307 Answers: 1 Comments: 0

∫(x^2 /(x^2 −(5/2)x−(3/2))) dx

$$\int\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{2}}{x}−\frac{\mathrm{3}}{\mathrm{2}}}\:{dx} \\ $$

Question Number 88286 Answers: 0 Comments: 1

Question Number 88253 Answers: 0 Comments: 0

∫ ((ln(x^2 +1) dx)/(x+1))

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

Question Number 88206 Answers: 1 Comments: 0

∫ ((x+x^3 )/(1+x^4 )) dx

$$\int\:\:\frac{\mathrm{x}+\mathrm{x}^{\mathrm{3}} }{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\:\mathrm{dx}\: \\ $$

Question Number 88197 Answers: 0 Comments: 0

prove that ∫_0 ^1 (√(((x^2 −2)/(x^2 −1)) ))dx=((π(√(2π)))/(Γ^2 ((1/4))))+((Γ^2 ((1/4)))/(4(√(2π))))

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}}\:}{dx}=\frac{\pi\sqrt{\mathrm{2}\pi}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}+\frac{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\mathrm{4}\sqrt{\mathrm{2}\pi}} \\ $$

Question Number 88196 Answers: 0 Comments: 0

Question Number 88194 Answers: 0 Comments: 4

find ∫(1/((√x)+(√(x+1))+(√(x+2)))) dx

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

Question Number 88181 Answers: 0 Comments: 1

Question Number 88177 Answers: 1 Comments: 0

prove that ∫_0 ^(2020) (x−[x])(√(x−[x])) dx=808

$${prove}\:{that} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2020}} \left({x}−\left[{x}\right]\right)\sqrt{{x}−\left[{x}\right]}\:{dx}=\mathrm{808} \\ $$

Question Number 88170 Answers: 0 Comments: 2

Prove that ∫_0 ^1 tcos nπtdt=(((−1)^n −1)/(n^2 π^2 ))

$${Prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {tcos}\:{n}\pi{tdt}=\frac{\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$

Question Number 88153 Answers: 0 Comments: 1

Question Number 88131 Answers: 1 Comments: 0

∫((x^2 +1)/(x−(√(1−2x))))dx

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

Question Number 88118 Answers: 1 Comments: 1

∫(dx/((x+1)×(√x)))

$$\int\frac{\mathrm{dx}}{\left(\mathrm{x}+\mathrm{1}\right)×\sqrt{\mathrm{x}}} \\ $$

Question Number 88104 Answers: 1 Comments: 2

∫_0 ^(+∞) (√(3+e^(−2x) ))dx

$$\int_{\mathrm{0}} ^{+\infty} \sqrt{\mathrm{3}+{e}^{−\mathrm{2}{x}} }{dx} \\ $$

Question Number 88097 Answers: 1 Comments: 0

∫(dx/((2x−3)^(2/3) ))

$$\int\frac{\mathrm{dx}}{\left(\mathrm{2x}−\mathrm{3}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} } \\ $$

Question Number 88089 Answers: 1 Comments: 1

∫(e^x /(e^2 −9))dx

$$\int\frac{{e}^{{x}} }{{e}^{\mathrm{2}} −\mathrm{9}}{dx} \\ $$

Question Number 88071 Answers: 1 Comments: 7

Question Number 88069 Answers: 1 Comments: 2

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