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IntegrationQuestion and Answers: Page 107

Question Number 133305 Answers: 3 Comments: 0

∫_0 ^( 2π) (dx/(5+3sin 2x)) =?

$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\frac{\mathrm{dx}}{\mathrm{5}+\mathrm{3sin}\:\mathrm{2x}}\:=? \\ $$

Question Number 133222 Answers: 1 Comments: 0

∫_0 ^1 ((x^3 dx)/((x−1)^3 +3x−5))

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{3}} \:\mathrm{dx}}{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{3}} +\mathrm{3x}−\mathrm{5}} \\ $$

Question Number 133268 Answers: 1 Comments: 0

....calculus... prove:: 𝛗=∫_(−∞) ^( +∞) (dx/((x^2 +π^2 )cosh(x)))=(4/𝛑) −1

$$\:\:\:\:\:\:\:\:\:\:\:\:....{calculus}... \\ $$$$\:\:{prove}:: \\ $$$$\:\:\:\boldsymbol{\phi}=\int_{−\infty} ^{\:+\infty} \frac{{dx}}{\left({x}^{\mathrm{2}} +\pi^{\mathrm{2}} \right){cosh}\left({x}\right)}=\frac{\mathrm{4}}{\boldsymbol{\pi}}\:−\mathrm{1} \\ $$$$ \\ $$

Question Number 133214 Answers: 0 Comments: 0

calculate f(ξ) =∫_0 ^∞ ((x sin(ξx))/(1+x^4 ))dx

$$\mathrm{calculate}\:\mathrm{f}\left(\xi\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{x}\:\mathrm{sin}\left(\xi\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 133213 Answers: 1 Comments: 0

calculate c(ξ) =∫_0 ^∞ ((cos(ξx))/(1+x^4 ))dx

$$\mathrm{calculate}\:\mathrm{c}\left(\xi\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{cos}\left(\xi\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 133228 Answers: 1 Comments: 0

....advanced calculus.... prove that :: Σ_(n=0) ^∞ ((Γ(n+(1/2))ψ(n+(1/2)))/(2^n .n!))=−(√(2π)) (γ+ln(2))....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{advanced}\:\:\:\:{calculus}.... \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\psi\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{2}^{{n}} .{n}!}=−\sqrt{\mathrm{2}\pi}\:\left(\gamma+{ln}\left(\mathrm{2}\right)\right).... \\ $$$$ \\ $$

Question Number 133187 Answers: 0 Comments: 0

.......CALCULUS...... lim _(n→∞) (n(ln(2)−Σ_(k=1) ^n (1/(n+k))))=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.......{CALCULUS}...... \\ $$$$\:\:\:\:\:{lim}\:_{{n}\rightarrow\infty} \left({n}\left({ln}\left(\mathrm{2}\right)−\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}+{k}}\right)\right)=? \\ $$$$ \\ $$

Question Number 133173 Answers: 3 Comments: 0

∫_0 ^( (1/2)) (√(1+(√(1−x^2 )))) dx =?

$$\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \:\sqrt{\mathrm{1}+\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=? \\ $$

Question Number 133125 Answers: 0 Comments: 1

find ∫_0 ^∞ ((cosx ch(2x))/((x^2 +x+3)^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cosx}\:{ch}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 133124 Answers: 0 Comments: 0

find ∫ (dx/((x+2)^2 (x^2 −x+1)^3 ))

$${find}\:\int\:\:\frac{{dx}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 133123 Answers: 1 Comments: 0

find ∫ ((x^2 dx)/(x^3 −2x+1))

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

Question Number 133121 Answers: 1 Comments: 0

let u_n =Σ_(k=1) ^n (1/(√k)) find a equivalent of u_n (n+→∞)

$${let}\:{u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{\sqrt{{k}}} \\ $$$${find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \left({n}+\rightarrow\infty\right) \\ $$

Question Number 133120 Answers: 0 Comments: 0

calculate A_n = ∫∫_([(1/n),n[) (e^(−x^2 −y^2 ) /(√(x^2 +y^2 +3)))dxdy and lim_(n→∞) A_n

$${calculate}\:{A}_{{n}} =\:\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\left[\right.\right.} \:\:\frac{{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } }{\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{3}}}{dxdy} \\ $$$${and}\:{lim}_{{n}\rightarrow\infty} {A}_{{n}} \\ $$

Question Number 133119 Answers: 2 Comments: 0

find ∫_0 ^∞ ((xsin(2x))/((x^2 +4)^3 ))dx

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

Question Number 133109 Answers: 2 Comments: 1

∫ (dx/((x^4 +1) ((x^4 +2))^(1/4) )) ?

$$\int\:\frac{{dx}}{\left({x}^{\mathrm{4}} +\mathrm{1}\right)\:\sqrt[{\mathrm{4}}]{{x}^{\mathrm{4}} +\mathrm{2}}}\:? \\ $$

Question Number 133107 Answers: 1 Comments: 0

... nice calculus... find:: 𝛗=^(???) ∫_0 ^( 1) (sin(x)+sin((1/x)))(dx/x)

$$\:\:\:\:\:\:\:\:...\:{nice}\:\:\:{calculus}... \\ $$$$\:\:\:{find}::\:\:\:\boldsymbol{\phi}\overset{???} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({sin}\left({x}\right)+{sin}\left(\frac{\mathrm{1}}{{x}}\right)\right)\frac{{dx}}{{x}} \\ $$

Question Number 133051 Answers: 0 Comments: 0

....advanced calculus.... evaluate: Σ_(n=1) ^∞ (H_n /(n^2 2^(n+1) ))=??

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{advanced}\:\:\:{calculus}.... \\ $$$$\:\:\:\:\:{evaluate}:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{H}_{{n}} }{{n}^{\mathrm{2}} \:\mathrm{2}^{{n}+\mathrm{1}} }=?? \\ $$

Question Number 133050 Answers: 2 Comments: 0

...nice ......calculus... 𝛗= ∫_(0 ) ^( 1) xli_3 (x)dx=???

$$\:\:\:\:\:\:\:\:\:\:\:...{nice}\:......{calculus}... \\ $$$$\:\:\:\:\:\boldsymbol{\phi}=\:\int_{\mathrm{0}\:} ^{\:\mathrm{1}} {xli}_{\mathrm{3}} \left({x}\right){dx}=??? \\ $$$$ \\ $$

Question Number 133048 Answers: 0 Comments: 0

nice .....calculus... evaluate ::Σ_(n=1) ^∞ ((H_n /(n^2 +n)))=?

$$\:\:\:\:\:\:\:{nice}\:.....{calculus}... \\ $$$$\:\:\:{evaluate}\:::\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{H}_{{n}} }{{n}^{\mathrm{2}} +{n}}\right)=? \\ $$$$ \\ $$

Question Number 133038 Answers: 1 Comments: 1

∫_0 ^1 ((x^4 (1−x)^4 )/(1+x^2 )) dx =?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{x}^{\mathrm{4}} \left(\mathrm{1}−{x}\right)^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:=? \\ $$

Question Number 133036 Answers: 2 Comments: 0

∫^( (π/2)) _0 (dx/(1+tan^(2014) (x))) = ((πe^q )/p) Find 2p−q.

$$\underset{\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \:\frac{{dx}}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{2014}} \left({x}\right)}\:=\:\frac{\pi\mathrm{e}^{\mathrm{q}} }{\mathrm{p}} \\ $$$$\mathrm{Find}\:\mathrm{2p}−\mathrm{q}.\: \\ $$

Question Number 133027 Answers: 4 Comments: 0

∫_0 ^2 x^5 (8−x^3 )^(1/3) dx

$$\overset{\mathrm{2}} {\int}_{\mathrm{0}} {x}^{\mathrm{5}} \left(\mathrm{8}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx} \\ $$

Question Number 133016 Answers: 3 Comments: 0

∫_0 ^(π/2) (tan(x))^(1/n) dx ...

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({tan}\left({x}\right)\right)^{\frac{\mathrm{1}}{{n}}} {dx}\:... \\ $$

Question Number 133004 Answers: 1 Comments: 0

If ∫ ((tan x)/(1+tan x+tan^2 x)) dx = x−(k/( (√A))) tan^(−1) (((k tan x+1)/( (√A))))+C where C is constant of integration. then the ordered pair (k,A) is equal to

$$\mathrm{If}\:\int\:\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{1}+\mathrm{tan}\:\mathrm{x}+\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}}\:\mathrm{dx}\:=\:\mathrm{x}−\frac{{k}}{\:\sqrt{{A}}}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{{k}\:\mathrm{tan}\:{x}+\mathrm{1}}{\:\sqrt{{A}}}\right)+\mathrm{C} \\ $$$$\mathrm{where}\:\mathrm{C}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{integration}. \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{ordered}\:\mathrm{pair}\:\left({k},\mathrm{A}\right)\:\mathrm{is}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\: \\ $$

Question Number 133001 Answers: 0 Comments: 1

Question Number 132987 Answers: 2 Comments: 0

....mathematical analysis... prove that:: 𝛗=∫_0 ^( ∞) ((sin^3 (x))/x^3 )dx=((3π)/8) ∗∗∗∗..........

$$\:\:\:\:\:\:\:\:\:\:\:\:\:....{mathematical}\:\:{analysis}... \\ $$$$\:{prove}\:\:{that}::\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{3}} \left({x}\right)}{{x}^{\mathrm{3}} }{dx}=\frac{\mathrm{3}\pi}{\mathrm{8}} \\ $$$$\:\:\:\:\ast\ast\ast\ast.......... \\ $$$$ \\ $$

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