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IntegrationQuestion and Answers: Page 120

Question Number 127616 Answers: 0 Comments: 1

If ∫_1 ^( 4) f(x) dx = 6 , then ∫_1 ^( 4) f(5−x) dx ?

$$\:\mathrm{If}\:\int_{\mathrm{1}} ^{\:\mathrm{4}} \mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{6}\:,\:\mathrm{then}\:\int_{\mathrm{1}} ^{\:\mathrm{4}} \mathrm{f}\left(\mathrm{5}−\mathrm{x}\right)\:\mathrm{dx}\:?\: \\ $$

Question Number 127605 Answers: 1 Comments: 0

find arg(z) where z=1+cosα+icosβ

$${find}\:{arg}\left({z}\right) \\ $$$${where}\:\boldsymbol{{z}}=\mathrm{1}+\boldsymbol{{cos}}\alpha+{icos}\beta \\ $$

Question Number 127604 Answers: 2 Comments: 0

Question Number 127587 Answers: 1 Comments: 0

Question Number 127575 Answers: 1 Comments: 0

I_n =∫_0 ^1 (1−t^2 )^n dt

$$\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{n}} \mathrm{dt} \\ $$

Question Number 127543 Answers: 2 Comments: 0

∫_0 ^1 (x^(2021) /(e^x −1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{2021}} }{\mathrm{e}^{\mathrm{x}} −\mathrm{1}}\mathrm{dx} \\ $$

Question Number 127539 Answers: 2 Comments: 0

...nice calculus... evaluate ::: Φ=∫_0 ^( ∞) ((ln(x))/((x^2 +1)^3 )) dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left({x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }\:{dx}=? \\ $$$$ \\ $$

Question Number 127528 Answers: 3 Comments: 0

∫_0 ^( π/4) ((sin x)/(sin x+cos x)) dx =?

$$\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:=?\: \\ $$

Question Number 127495 Answers: 1 Comments: 0

Question Number 127468 Answers: 3 Comments: 1

∫_0 ^( ∞) ((x^3 sin (λx))/(x^4 +4)) dx =?

$$\:\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{x}^{\mathrm{3}} \:\mathrm{sin}\:\left(\lambda\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} +\mathrm{4}}\:\mathrm{dx}\:=?\: \\ $$

Question Number 127446 Answers: 1 Comments: 0

...challanging integral... prove that :: Ω=∫_0 ^( ∞) (cos(x)−(1/(1+x^2 )))(dx/x) = −γ

$$\:\:\:\:\:\:\:\:\:\:...{challanging}\:\:{integral}... \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \left({cos}\left({x}\right)−\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} \:}\right)\frac{{dx}}{{x}}\:=\:−\gamma\:\: \\ $$$$ \\ $$

Question Number 127432 Answers: 1 Comments: 0

...CALCULUS... ∅=∫_(−∞) ^( +∞) cos(((πx^2 )/2))dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\mathrm{CALCULUS}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\varnothing=\int_{−\infty} ^{\:+\infty} {cos}\left(\frac{\pi{x}^{\mathrm{2}} }{\mathrm{2}}\right){dx}=? \\ $$$$ \\ $$

Question Number 127420 Answers: 1 Comments: 0

∫_0 ^( ∞) ((x−1)/( (√(2^x −1)) ln (2^x −1))) dx ?

$$\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{x}−\mathrm{1}}{\:\sqrt{\mathrm{2}^{{x}} −\mathrm{1}}\:\mathrm{ln}\:\left(\mathrm{2}^{{x}} −\mathrm{1}\right)}\:{dx}\:? \\ $$

Question Number 127405 Answers: 0 Comments: 0

μ = ∫ (x^7 /(x^(10) +169)) dx

$$\:\mu\:=\:\int\:\frac{{x}^{\mathrm{7}} }{{x}^{\mathrm{10}} +\mathrm{169}}\:{dx}\: \\ $$

Question Number 127377 Answers: 0 Comments: 2

Question Number 127370 Answers: 1 Comments: 1

... advanced calculus .. prove:: ((1023)/(134))∫_0 ^( ∞) ((x^(2/5) +x^((−2)/5) )/((1+x^2 )(1+1024x^2 )))dx=(π/ϕ) ϕ: golden ratio...

$$\:\:\:\:\:\:\:\:\:...\:\:{advanced}\:\:{calculus}\:\:.. \\ $$$$\:\:{prove}:: \\ $$$$\:\:\:\frac{\mathrm{1023}}{\mathrm{134}}\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{\frac{\mathrm{2}}{\mathrm{5}}} +{x}^{\frac{−\mathrm{2}}{\mathrm{5}}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{1024}{x}^{\mathrm{2}} \right)}{dx}=\frac{\pi}{\varphi} \\ $$$$\:\:\:\varphi:\:{golden}\:\:{ratio}... \\ $$$$ \\ $$

Question Number 127368 Answers: 1 Comments: 0

∫ ((√(tan x))/( (√(tan^2 x − 1)))) dx

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

Question Number 127355 Answers: 1 Comments: 0

∫ (dx/((1+x^2 )(√(1−x^2 )))) =?

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

Question Number 127350 Answers: 1 Comments: 0

∫ (x^2 /((x^2 −1)^(5/2) )) dx ?

$$\:\int\:\frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{5}}{\mathrm{2}}} }\:{dx}\:? \\ $$

Question Number 127349 Answers: 1 Comments: 1

∫ (((1−s^2 )^(5/2) )/s^8 ) ds =?

$$\:\int\:\frac{\left(\mathrm{1}−{s}^{\mathrm{2}} \right)^{\mathrm{5}/\mathrm{2}} }{{s}^{\mathrm{8}} }\:{ds}\:=? \\ $$

Question Number 127435 Answers: 0 Comments: 0

Question Number 127341 Answers: 0 Comments: 0

Question Number 127464 Answers: 1 Comments: 0

Question Number 127237 Answers: 5 Comments: 1

Nice...∫ ((√(1−ln^2 (x)))/(x ln (x))) dx ∫ (√(x/(1−x^3 ))) dx ∫ (√((4−x)/x)) dx

$$\:{Nice}...\int\:\frac{\sqrt{\mathrm{1}−\mathrm{ln}\:^{\mathrm{2}} \left({x}\right)}}{{x}\:\mathrm{ln}\:\left({x}\right)}\:{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\sqrt{\frac{{x}}{\mathrm{1}−{x}^{\mathrm{3}} }}\:{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\sqrt{\frac{\mathrm{4}−{x}}{{x}}}\:{dx}\: \\ $$

Question Number 127236 Answers: 1 Comments: 0

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

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

Question Number 127224 Answers: 2 Comments: 0

... calculus (I) −complex analysis... calculate :: Φ = ∫_0 ^( ∞) ((ln(x))/(x^2 +3x+2)) dx=((ln^2 (2))/2)

$$\:\:...\:{calculus}\:\:\left({I}\right)\:−{complex}\:{analysis}... \\ $$$$\:\:\:\:{calculate}\:::\: \\ $$$$\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}}\:{dx}=\frac{{ln}^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

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