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IntegrationQuestion and Answers: Page 131

Question Number 115051 Answers: 2 Comments: 6

∫x^2 (√(x^2 −2))dx

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

Question Number 115030 Answers: 2 Comments: 0

∫_(−(π/2)) ^(π/2) (√(sec x−cos x)) dx =?

$$\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\sqrt{\mathrm{sec}\:{x}−\mathrm{cos}\:{x}}\:{dx}\:=? \\ $$

Question Number 115026 Answers: 2 Comments: 0

Solve: ∫_(1/π) ^(1/2) ln ⌊(1/x)⌋dx

$$\mathrm{Solve}:\:\:\int_{\mathrm{1}/\pi} ^{\mathrm{1}/\mathrm{2}} \mathrm{ln}\:\lfloor\frac{\mathrm{1}}{{x}}\rfloor{dx} \\ $$

Question Number 115009 Answers: 2 Comments: 0

∫_C (e^z /(1−cos z))dz ; C:∣z∣=1

$$\int_{\mathrm{C}} \frac{{e}^{{z}} }{\mathrm{1}−\mathrm{cos}\:{z}}{dz}\:;\:\mathrm{C}:\mid{z}\mid=\mathrm{1} \\ $$

Question Number 115000 Answers: 2 Comments: 0

....nice math... if y =(cos(2x))^(−(1/2)) then prove :: y+y^(′′) = 3y^5 ...m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:{math}... \\ $$$$ \\ $$$$\:{if}\:\:{y}\:=\left({cos}\left(\mathrm{2}{x}\right)\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} \:{then} \\ $$$${prove}\:::\:\:\:{y}+{y}^{''} =\:\mathrm{3}{y}^{\mathrm{5}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}... \\ $$

Question Number 114933 Answers: 2 Comments: 0

find the value of ∫_0 ^∞ ((cos(2x))/(x^4 +x^2 +1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 114880 Answers: 1 Comments: 1

∫ln (sin (x))dx=?

$$\int\mathrm{ln}\:\left(\mathrm{sin}\:\left({x}\right)\right){dx}=? \\ $$

Question Number 114773 Answers: 1 Comments: 0

Evaluate: ∫ (1/(sin^5 x + cos^5 x)) dx

$$\: \\ $$$$\:\:\mathrm{Evaluate}:\:\:\int\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{5}} \mathrm{x}\:+\:\mathrm{cos}^{\mathrm{5}} \mathrm{x}}\:\mathrm{dx} \\ $$

Question Number 114699 Answers: 1 Comments: 1

∫_0 ^(π/2) (dx/( (√(1+tan^4 x)))) ?

$$\:\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{{dx}}{\:\sqrt{\mathrm{1}+\mathrm{tan}\:^{\mathrm{4}} {x}}}\:? \\ $$

Question Number 114681 Answers: 3 Comments: 0

Prove that ∫_0 ^∞ (x^n /(e^x −1))dx=n!ζ(n+1)

$${Prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{n}} }{{e}^{{x}} −\mathrm{1}}{dx}={n}!\zeta\left({n}+\mathrm{1}\right) \\ $$

Question Number 114671 Answers: 1 Comments: 0

Integrate ∫ ((x^4 +x^(−4) )/(x^6 +x^(−6) ))dx

$$ \\ $$$$\:\mathrm{Integrate}\:\:\int\:\frac{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{−\mathrm{4}} }{\mathrm{x}^{\mathrm{6}} +\mathrm{x}^{−\mathrm{6}} }\mathrm{dx} \\ $$

Question Number 114635 Answers: 1 Comments: 3

find ∫_0 ^1 x^4 ln(x)ln(1−x^2 )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{4}} \mathrm{ln}\left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}\:\: \\ $$

Question Number 114566 Answers: 2 Comments: 0

Question Number 114541 Answers: 1 Comments: 0

Question Number 114526 Answers: 1 Comments: 2

(1/π)∫_0 ^π e^(2cos θ) dθ ?

$$\frac{\mathrm{1}}{\pi}\underset{\mathrm{0}} {\overset{\pi} {\int}}\:{e}^{\mathrm{2cos}\:\theta} \:{d}\theta\:? \\ $$

Question Number 114472 Answers: 0 Comments: 2

... calculus... evaluate :: I=∫_0 ^( (π/2)) ((tan(2x))/( (√(sin^4 (x)+4cos^2 (x)))−(√(cos^4 (x)+4sin^2 (x) )))) dx= ??? ...m.n.july.1970....

$$\:\:\:\:\:\:\:\:\:...\:\:{calculus}... \\ $$$${evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{tan}\left(\mathrm{2}{x}\right)}{\:\sqrt{{sin}^{\mathrm{4}} \left({x}\right)+\mathrm{4}{cos}^{\mathrm{2}} \left({x}\right)}−\sqrt{{cos}^{\mathrm{4}} \left({x}\right)+\mathrm{4}{sin}^{\mathrm{2}} \left({x}\right)\:}}\:{dx}=\:??? \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:...{m}.{n}.{july}.\mathrm{1970}.... \\ $$$$ \\ $$

Question Number 114467 Answers: 1 Comments: 0

∫x sin^n (x) dx

$$\int{x}\:{sin}^{{n}} \left({x}\right)\:{dx} \\ $$

Question Number 114403 Answers: 1 Comments: 1

∫ ((ln (1+x^4 ))/x) dx

$$\:\:\:\:\:\:\:\:\:\int\:\frac{\mathrm{ln}\:\left(\mathrm{1}+{x}^{\mathrm{4}} \right)}{{x}}\:{dx} \\ $$$$ \\ $$

Question Number 114395 Answers: 1 Comments: 0

find ∫_0 ^∞ (((1+x)^(−(3/4)) −(1+x)^(−(1/4)) )/x)dx

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

Question Number 114330 Answers: 2 Comments: 1

Question Number 114320 Answers: 0 Comments: 0

Question Number 114302 Answers: 6 Comments: 2

.... calculus .... evaluate ::: i::∫_0 ^( 1) t^2 ln(t)ln(1−t)dt=??? ii::: ψ^′ ((1/4))=??? iii::: ∫_0 ^(π/8) ln(tan(x))dx =???

$$\:\:\:\:\:\:\:\:....\:{calculus}\:.... \\ $$$$\:\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${i}::\int_{\mathrm{0}} ^{\:\mathrm{1}} {t}^{\mathrm{2}} {ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right){dt}=??? \\ $$$${ii}:::\:\psi^{'} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)=??? \\ $$$${iii}:::\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} {ln}\left({tan}\left({x}\right)\right){dx}\:=??? \\ $$$$ \\ $$

Question Number 114161 Answers: 2 Comments: 0

∫ (dx/(x^4 −5x^2 −16))

$$\:\int\:\frac{{dx}}{{x}^{\mathrm{4}} −\mathrm{5}{x}^{\mathrm{2}} −\mathrm{16}} \\ $$

Question Number 114146 Answers: 0 Comments: 2

prove ∫_0 ^1 ((t^(n+2) φ(t,1,n+2)+ln(1−t)+t H_(n+1) )/(t(t−1)))dt =((H_(n+1) ^((2)) −(H_n )^2 )/2)

$${prove} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{t}^{{n}+\mathrm{2}} \phi\left({t},\mathrm{1},{n}+\mathrm{2}\right)+{ln}\left(\mathrm{1}−{t}\right)+{t}\:{H}_{{n}+\mathrm{1}} }{{t}\left({t}−\mathrm{1}\right)}{dt} \\ $$$$=\frac{{H}_{{n}+\mathrm{1}} ^{\left(\mathrm{2}\right)} −\left({H}_{{n}} \right)^{\mathrm{2}} }{\mathrm{2}} \\ $$

Question Number 114135 Answers: 2 Comments: 0

....Advanced mathematics ... i:: prove that : Ω=(1/π)∫_0 ^( ∞) (1/((x^2 −x+1)^2 (√x)))dx =1 ii::evaluate :: Φ = ∫_0 ^( 1) x^2 ln(x) ln(1−x)dx=??? ....m.n.july. 1970....

$$\:\:\:\:\:\:\:\:\:\:\:\:....\mathscr{A}{dvanced}\:\:{mathematics}\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:{prove}\:\:{that}\::\:\:\:\:\Omega=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} \sqrt{{x}}}{dx}\:=\mathrm{1}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::{evaluate}\:::\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}} \:{ln}\left({x}\right)\:{ln}\left(\mathrm{1}−{x}\right){dx}=??? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{m}.{n}.{july}.\:\mathrm{1970}.... \\ $$$$\: \\ $$

Question Number 114105 Answers: 0 Comments: 1

∫ ((arctan(e^x ))/( (√x))) dx

$$\int\:\frac{{arctan}\left({e}^{{x}} \right)}{\:\sqrt{{x}}}\:{dx} \\ $$

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