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IntegrationQuestion and Answers: Page 120

Question Number 127732 Answers: 0 Comments: 0

z=x+iy why ((f(z))/(z−a)) not analytical? / not analytical at z=a?

$${z}={x}+{iy} \\ $$$${why}\:\frac{{f}\left({z}\right)}{{z}−{a}}\:{not}\:{analytical}?\:/\:{not}\:{analytical}\:{at}\:{z}={a}? \\ $$$$ \\ $$

Question Number 127704 Answers: 1 Comments: 0

if f(x)= { ((x−n ; 2n ≤ x ≤2n+1)),((n+1 ; 2n+1≤x≤2n+2 )) :} where n =0,1,2,3,..,9 find ∫_0 ^(20) f(x)dx

$$ \\ $$$${if}\:{f}\left({x}\right)=\begin{cases}{{x}−{n}\:;\:\mathrm{2}{n}\:\leqslant\:{x}\:\leqslant\mathrm{2}{n}+\mathrm{1}}\\{{n}+\mathrm{1}\:;\:\mathrm{2}{n}+\mathrm{1}\leqslant{x}\leqslant\mathrm{2}{n}+\mathrm{2}\:}\end{cases}\:{where}\:\:{n}\:=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},..,\mathrm{9} \\ $$$${find}\:\int_{\mathrm{0}} ^{\mathrm{20}} {f}\left({x}\right){dx} \\ $$

Question Number 127679 Answers: 0 Comments: 0

its 9:30pm in Cameroon

$${its}\:\mathrm{9}:\mathrm{30}{pm}\:{in}\:{Cameroon} \\ $$

Question Number 127631 Answers: 0 Comments: 0

Let f∈C^∞ (R,R) , ∀ n∈N M_n =∣∣f^((n)) ∣∣_∞ and u_n =((2^(n−1) M_n )/M_(n−1) ) for n≥1 Show that if M_1 <(√(2M_0 M_2 )) then u_n <u_(n+1) for n≥1

$${Let}\:{f}\in{C}^{\infty} \left(\mathbb{R},\mathbb{R}\right)\:,\:\forall\:{n}\in\mathbb{N}\:\:\:{M}_{{n}} =\mid\mid{f}^{\left({n}\right)} \mid\mid_{\infty} \:\: \\ $$$${and}\:\:{u}_{{n}} =\frac{\mathrm{2}^{{n}−\mathrm{1}} {M}_{{n}} }{{M}_{{n}−\mathrm{1}} }\:\:\:{for}\:{n}\geqslant\mathrm{1}\: \\ $$$${Show}\:{that}\:{if}\:\:\:{M}_{\mathrm{1}} <\sqrt{\mathrm{2}{M}_{\mathrm{0}} {M}_{\mathrm{2}} }\:{then}\:{u}_{{n}} <{u}_{{n}+\mathrm{1}} \:{for}\:{n}\geqslant\mathrm{1} \\ $$

Question Number 127618 Answers: 2 Comments: 0

∫ (dx/(x^(2021) (1+x^(2020) ))) ?

$$\:\int\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2021}} \left(\mathrm{1}+\mathrm{x}^{\mathrm{2020}} \right)}\:?\: \\ $$

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

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